UC-NRLF 


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&  4S?  J^L^cr*^ 


TREATISE 


CONSTRUCTION,  PROPERTIES,  AND  ANALOGIES 


THREE    CONIC    SECTIONS 


BY  THE 

Rev.  B.  BRIDGE,  B.  D.  F.  R.  S. 

FELLOW  OF  ST.  PETER'S  COLLEGE,  CAMBRIDGE. 


FROM  THE  SECOND  LONDON  EDITION, 

WITH  ADDITIONS  AND  ALTERATIONS  BY  THE  AMERICAN  EDITOR. 


NEW  HAVEN: 

DURRIE  AND  PECK. 

NEW-YORK  : COLLINS,  KEESE,  AND  CO, 

1839. 


t  \ 


Entered  according  to  the  Act  of  Congress,  in  the  year  1831, 

by  Hezekiaii  Howe, 

in  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


STEREOTYPED  BY 

FRANCIS   F.   RIPLEY, 

New  Yokk. 


ADVERTISEMENT, 


The  present  edition  of  Bridge's  Conic  Sections,  is  reprinted,  with 
a  few  alterations,  from  the  second  London  edition.  Such  changes 
only  have  been  made,  as  seemed  necessary  to  adapt  the  work  to  the 
purpose  for  which  it  is.  intended,  namely,  that  of  furnishing  a  text- 
book for  recitation  in  Colleges.  To  this  end,  the  propositions  have 
been  enunciated  without  the  use  of  letters  and  without  reference  to 
particular  diagrams.  As  it  is  true,  however,  that  a  proposition  is 
more  readily  comprehended,  when  it  is  asserted  of  the  lines  and  an- 
gles of  some  particular  figure,  immediately  before  the  eye,  than  when 
expressed  in  general  terms ;  it  has  been  thought  proper  to  introduce 
the  general  enunciations  at  the  close  of  the  demonstrations  to  which 
they  belong,  and  to  leave  the  author's  statements  at  the  commence- 
ment of  those  demonstrations  unaltered.  In  this  respect  the  conven- 
ience of  students  has  been  consulted,  rather  than  the  usual  practice 
of  writers. 

The  demonstrations,  as  they  stand  in  the  original  work  of  Bridge, 
are  in  general  so  much  distinguished  for  conciseness 'and  simplicity, 
and  leave  so  little  to  be  supplied  by  the  student,  (a  circumstance  of 
great  importance  in  a  book  designed  for  large  classes,)  that  it  has  been 
thought  best  to  vary  from  the  author  only  in  a  very  few  instances. 
Alterations  have,  however,  been  made,  when  they  seemed  likely  to 
be  attended  by  any  material  advantage. 

The  number  of  the  propositions  has  been  somewhat  increased,  not 
for  the  purpose  of  completing  the  enumeration  of  particular  properties 
of  the  Conic  Sections,  which,  in  a  work  like  this,  considering  the  fer- 
tility of  the  subject,  would  be  equally  impracticable  and  useless  ;  but 
in  order  to  exhibit  to  the  student  how  far  many  truths  may  be  gener- 
alized, which  he  is  apt  to  consider  as  limited  by  particular  circum- 
stances.    A  few  propositions,  not  of  this  kind,  have  been  added,  as 


4  ADVERTISEMENT. 

being  among  the  more  curious  of  those  which  Bridge  has  omitted 
to  notice. 

For  the  convenience  of  students,  some  references,  particularly  to- 
wards the  close  of  the  book,  have  been  made  to  the  mathematical 
treatises  of  President  Day. 

The  original  numbering  of  the  Properties  and  of  the  Articles  has 
been  suffered  to  stand  ;  and  whenever  any  thing  has  been  inserted  in 
the  body  of  the  work,  the  number  of  the  preceding  article  has  been 
repeated  with  a  letter  annexed.  The  additional  Properties  are  dis- 
tinguished by  the  capitals  A,  B,  C,  &c.  A  few  notes  contain  what- 
ever else  is  peculiar  to  this  Edition. 

F.  A.  P.  BARNARD. 

Yale  College,  June  20,  1831. 


CONTENTS. 


CHAPTER  I. 

Introduction. 

Page. 

Sect.  I.  On  the  nature  of  the  Curves  arising  from  the  cutting  of  a 

Cone  obliquely  to  its  base,  .....       9 

II.  On  the  mode  of  describing  the  Conic   Sections  upon  a 

plane, 12 


CHAPTER  II. 

On  the  Parabola. 

III.  Definitions, .     18 

IV.  On  the  Properties  of  the  Parabola,       ....  19 

CHAPTER  III. 

On  the  Ellipse. 

V.  Definitions, 34 

VI.  On  the  Properties,  of  the  Ellipse,  ....  35 

CHAPTER  IV. 

On  the  Hyperbola. 
VII.  Definitions,      . 57 

VHI.  Properties  of  the  Hyperbola  analogous  to  those  of  the 

Ellipse, 59 


CONTENTS. 


Page. 


IX.  On  the  Properties  of  the  Hyperbola  derived  from  its  re- 
lation to  the  Asymptote, 73 

X.  On  the  Properties  of  the  Equilateral  Hyperbola,      .         81 


CHAPTER  V. 

On  the  Curvature  of  the  Conic  Sections. 

XI.  On  Curvature,  and  the  Variation  of  Curvature,  •  .85 

XII.  On  the  Curvature  of  the  Parabola,  •  92 

XIII.  On  the  Curvature  of  the  Ellipse, 94 

m 

XIV.  On  the  Curvature  of  the  Hyperbola,  ...  97 

CHAPTER  VI. 

On  the  analogous  Properties  of  the  Three  Conic  Sections. 

XV.  On  the  changes  which  take  place  in  the  nature  of  the 
*  Curve  described  upon  the  surface  of  a  Cone,  during  the 
revolution  of  the  plane  of  intersection,  .         .         .99 

XVI.  On  the  mode  of  constructing  the  Three  Conic  Sections  by 
means  of  a  Directrix,  and  the  Properties  derived 
therefrom,     .         . 102 

XVII.  On  the  analogous  properties  of  the  Normal,  Latus-rec- 
tum,  Radius  of  Curvature,  &c.  &c.  in  all  the  Conic 
Sections, Ill 

CHAPTER  VII. 

On  the  method  of  finding  the  dimensions  of  Conic  Sections  whose 
Later a-recta  are  given,   and  of  describing  such  as  shall  pass, 
through  certain  given  points. 

XVIIL  On  the  method  of  finding  the  dimensions  of  Conic  Sec- 
tions, whose  latera-recta  are  given, 117 


CONTENTS.  7 

Page. 

XIX.  On  the  method  of  describing  Conic  Sections  which  shall 

pass  through  three  given  points, 121 

CHAPTER  VIII. 

On  the  Quadrature  of  the  Conic  Sections. 

XX.  On  the  relation  which  obtains  between  the  areas  of  Conic 
Sections  of  the  same  kind,  having  the  same  vertex  and 
axis ;  and  on  the  Quadrature  of  the  Parabola,  Ellipse, 
and  Hyperbola, 126 

XXI.  On  the  Quadrature  of  the  Parabola,  according  to  the 

method  of  the  Ancients, 133 


CONIC   SECTIONS. 


CHAPTER  I. 


INTRODUCTION. 

A  cone  is  a  solid  figure  formed  by  the  revolution  of  a  right  an- 
gled triangle  about  one  of  its  sides.  (Euc.  Def.  11.  3.  Sup.)  From 
the  manner  in  which  this  solid  is  generated,  it  is  evident  that  if  it  be 
cut  by  a  plane  parallel  to  its  base,  the  intersection  of  the  plane  with 
the  solid,  will  be  a  circle,  since  this  section  will  coincide  with  the 
revolution  of  a  perpendicular  to  the  fixed  side  of  the  triangle ;  and 
if  it  be  cut  by  a  plane  passing  through  its  vertex,  the  intersection  will 
be  a  triangle,  the  sides  of  which  will  correspond  to  the  hypothenuse 
of  the  generating  triangle,  in  different  positions,  or  at  different  periods 
of  the  revolution.  If  the  plane  by  which  the  cone  is  cut  be  not  par- 
allel to  the  base,  or  do  not  pass  through  the  vertex,  then  the  line  tra- 
ced out  upon  its  surface  will  be  one  of  those  curves  more  particular- 
ly distinguished  by  the  name  of  Conic  Sections,  the  properties  of 
which  are  to  be  made  the  subject  of  the  following  Treatise. 


(1.)  Let  BEFGp  be  a  cone,  and  let  it  be  cut  by  a  plane  EEnG 
perpendicular  to  its  base  and  passing  through  its  vertex ;  then  the 
section  BEG  will  be  a  triangle.  Next,  let  it  be  cut  by  a  plane  pAon 
at  right  angles  to  the  plane  BEwG,  and  parallel  to  a  plane  touching 
the  side  BE  of  the  cone  ;  then  the  curve  line  pPAOo,  which  is  form- 
ed by  the  intersection  of  this  latter  plane  with  the  surface  of  the 
cone,  is  called  a  Parabola. 

C.S  2 


10 


INTRODUCTION. 


For  the  purpose  of  investigating  the  nature  of  this  curve,  let 
CPDON  be  a  plane  parallel  to  the  base  of  the  cone  ;  the  intersection 
CPDO  of  this  plane  with  the 
cone  will  be  a  circle.  Since  the 
plane  BEnG  divides  the  cone  in- 
to two  equal  parts,  CD  (the  com- 
mon intersection  of  the  planes 
BEnG,  CPDON)  will  be  *  di- 
ameter of  that  circle ;  and  for 
the  same  reason  EG  will  be  a 
diameter  of  the  circle  EpGoF. 
Let  ANn  be  the  common  inter- 
section of  the  planes  BEnG, 
p Aon,  and  PNO,  pno,  those  of 
the  plane  pAon,  with  the  planes 
CPDON,  E^GoF  respective- 
ly. Because  the  planes  pAon,  p  0 
CPDON  are  perpendicular  to  the  plane  BEnG,  PNO  must  be 
perpendicular  to  the  plane  BEnG,  (Euc.  18.  2.  Sup.)  and  conse- 
quently perpendicular  to  the  two  lines  AN,  ND  drawn  in  that  plane  ; 
(Euc.  Def.  1.  2.  Sup.)  for  the  same  reason  pno  is  perpendicular  to 
the  two  lines  An,  nG.     Hence  by  the  property  of  the  circle  CNx 


PN2 


pit- 


ND=PN2,  or  ND=TW  f  and  EnxnG=pn2,  or  nG=%— 


CN 


En 


Now  since  An  is  parallel  to  BE,  and  CD  parallel  to  EG,  the  fig- 
ure CNnE  is  a  parallelogram;    .\CN=E?&.     By  similar  triangles 

AND,  AnG ;   AN  :  An  : :  ND  :  nG  : :  ^J  ;?£l : :  (since  CN=En) 

CN     En      v  ' 

PN2:pn2. 


(2.)  Hence  the  nature  of  the  curve  APp  is  such,  that  if  it  begins 
to  be  generated  from  the  given  point  A,  and  PN  is  drawn  always  at 
right  angles  to  AN,  AN  will  vary  as  PN2.  And  the  same  may  be 
said  with  respect  to  the  relation  of  AN  and  NO  on  the  other  side  of 
ANn. 


INTRODUCTION. 


11 


(3.)  Next,  let  the  plane  MPAoM  be  drawn,  as  before,  perpen- 
dicular to  the  plane  BEG,  but  passing  through  the  sides  of  the  cone 
BE,  BG ;  then  the  curve  MPAoM,  formed  by  the  intersection  of 
this  plane  with  the  surface  of  the  cone,  is  called  an  Ellipse. 


In  this  case,  draw  two  planes,  CPDON,  UpKon,  parallel  to  the 
base  of  the  cone  ;  then,  for  the  same  reason  as  before,  PN  will  be 
perpendicular  both  to  AN  and  ND,  and  pn  will  be  perpendicular 
both  to  An  and  nK  ;  .-.  NCxND=PN2,  and  rcHxrcK=^w2. 


By  sim.  triangles  AND,  AriK ; 


we  have 


AN 
NM 


ANxNM 


An 
rcM 


AnxnM. 


MNC,  JVbiH; 
ND        :  nK, 
NC         : nR ; 


NCxND  :  nUxnK 
PN2       :  pri*. 


(4.)  The  nature  of  the  curve  APM  therefore  is  such,  that  if  A  and 
M  are  given  points,  and  PN  be  always  drawn  at  right  angles  to  AM 
between  the  points  A,  M,  ANxNM  will  vary  as  PN2 ;  and  the  same 
with  respect  to  the  relation  between  ANxNM  and  NO2. 


12 


INTRODUCTION. 


(5.)  Lastly,  let  the  plane  pAon  be  drawn,  as  before,  perpendicu- 


lar to  the  plane  BEG,  but 
cutting  the  side  BG  in  A, 
and,  when  produced,  meet- 
ing a  plane  drawn  touching 
the  other  side  EB  produ- 
ced,  in  M  ;  then  the  curve 
pVAOo  formed  by  the  in- 
tersection of  the  plane  pAon 
with  the  surface  of  the  cone, 
is  called  an  Hyperbola. 

Let  the  plane  CPDON  be 
drawn  parallel  to  the  base ; 
then,  by  similar    triangles, 


AND,  AnG  ; 

we  have 


MNC,  MrcE ;    F:( 


c/^       /\ 

N>\  D 

-""rX 

/        JLL— 

ps               \v 

n       \    ^X 

AN 

NM 


An 

wM 


ND 

NC 


wG, 
wE; 


ANxNM 


AwxnM  : 


NDxNC  :  nGxnE  : :  PN2 :  pn\ 


(6.)  Hence  the  nature  of  the  curve  AVp  is  such,  that  if  A  and 
M  are  given  points,  and  PN  be  always  drawn  at  right  angles  to  AN, 
the  point  A  lying  between  M  and  N,  then  ANxNM  will  vary  as 
PN2 ;  and  the  same  with  respect  to  the  relation  of  ANxNM  and 
NO2. 


II. 


Having  thus  explained  the  nature  of  the  curves  arising  from  the 
intersection  of  a  plane  with  the  surface  of  a  cone,  we  now  proceed 
to  show  how  these  curves  may  be  constructed  geometrically. 


INTRODUCTION. 


13 


(7.)  Let  ELF  be  a  line  given  in  position,  and  LZ  another  line 
drawn  at  right  angles  to  it  in  the  point  L.     In  LZ  take  any  point 


S,  and  bisect  SL  in  A.  Let  a  point  P  move  from  A,  in  such  a 
manner  that  it  may  always  be  at  equal  distances  from  S  and  the  line 
ELF  (or,  in  other  words,  let  the  line  SP  revolve  round  S  as  a  cen- 
ter, and  intersect  another  line  PM  moving  parallel  to  LZ,  in  such 
a  manner  that  SP  may  be  always  equal  to  PM  ;)  then  the  point  P 
will  trace  out  a  curve  OAP,  having  two  similar  branches,  AP,  AO, 
one  on  each  side  of  the  line  AZ  ;  which  curve  will  be  a  Parabola. 


To  show  that  this  curve  will  be  a  parabola^  draw  PNO  at  right 
angles  to  AZ  ;  then  LNPM  will  be  a  parallelogram,  and  LN=PM 
=SP ;  but  LN=AN+AL=AN+AS  (since  AL=AS  by  construc- 
tion,) .-.SP=AN+AS. 


Let  AN=:r, 

PN=y, 

SA=a; 
thenSP=AN+AS 

=x-\-a, 
and  SN=AN— -AS, 
=x — a. 


Now  PN2=SP2— SN2,  (Euc.  47.  1.) 
or  y2=(#-fa)2— (x— a)2 

* x*+2ax+a2— x*+2ax— a8 


14 


INTRODUCTION. 


Since  Aa  is  a  constant  quantity,  x  varies  as  y2,  or  AN  ccPN2 ; 
the  relation  between  AN  and  PN  is  therefore  the  same  as  in  Art.  2. ; 
hence  the  curve  AP  is  a  Parabola.* 

(8.)  Next,  take  any  line  SH,  and  produce  it  both  ways  towards 
A  and  M.  Let  a  point  P  begin  to  move  from  A,  in  such  a  manner 
that  the  sum  of  its  distances  from  S  and  H  may  be  always  the  same, 
(or,  in  other  words,  let  two  lines,  SP,  PH,  intersecting  each  other  in 
P,  revolve  round  the  fixed  points  S  and  H,  in  such  a  manner  that 
SP+PH  may  be  a  constant  quantity ;)  then  the  curve  APMO  tra- 
ced out  by  the  point  P  will  be  an  Ellipse. 

To  prove  this,  it  may  be  observed  that  when  P  is  at  A,  then 
HA-f  AS  or  HS-f-2AS,  is  equal  to  that  constant  quantity ;  and 
when  P  is  at  M,  SM-fMH  or  HS+2HM,  is  equal  to  the  same 


quantity.  Hence  HS+2AS=HS+2HM,  from  which  it  appears 
that  2AS  =  2HM,  or  AS  =  HM.  Now  SP+PH  =  HA+AS= 
HA+HM=AM  ;  bisect  therefore  SH  in  C,  and  make  CM  equal  to 


*  Geometrically  demonstrated  thus  : 
Since  SP=AN+AS,  SP2=(AN+AS)2=  (Euc.  8.  2.)  SN2+4AS.AN. 
But  (Euc.  47.  1.)  SP2=SN2+PN2 ;  .-.  SN2+PN2=SN2-f4AS.AN,  or 
PN2=4AS.AN. 


INTRODUCTION.  15 

CA,  then  M  will  be  the  point  where  the  curve  cuts  the  line  SH 
produced  ;  and  AM  will  be  the  constant  quantity  to  which  SP-f  PH 
is  equal. 


Let  Then 


AC  or  CM=a, 
SC  or  CH=6, 


and  SP+PH=AM= 
2AC=2a;    if  .-.SP=a— z, 

HP  will  be  equal  to  a+z. 


AN=AC-CN=a— *, 

NM=CM+CN=a+^ 

CN=-r,  [-SN  -SC—  CN=6— x, 

and  PN=y;  j  NH  =CH+ CN=6 -{-#, 

Draw  PNO  at  right  angles  to  AM,  then  (Euclid,  47.  1.)  we  have, 

HP2=PN2+NH*,  or  (a+z)*  =y2  +(b+x)2 ,     (A) 

and   SP2=PN2-j-NS2,  or  (a— zy=y*+(b— x)\     (B) 

bx 
Subtract  (B)  from  (A),  then  4.az=4.bx,  or  #=— ;  substitute  this  value 

for  z  in  equation  (A),  and  it  becomes 

(bx\  2 

which  reduced  is 

a 4  +2a2^+62^2=a2y2 +a?b2+2a'ibx+a'ix*, 

or  a4—  a262— a2^2+63^=a2y2, 

i.e.  (a2— 62)x(«2— #2)=a2y2. 
But  since  a  and  6  are  constant  quantities,  a2 — #2  varies  as  y2 ;  now 
a*—a;*=(a—x)  X (a+#) ;  .-.  (a—x)x(a+x)  ocy2,  or  ANxNM  ocPN2 ; 
hence,  as  N  lies  between  A  and  M,  the  relation  between  ANxNM 
and  PN2  is  such,  that  the  curve  APM  is  an  Ellipse. 

(9.)  Lastly,  Take  any  line  SH,  and  let  the  two  lines  SP,  HP,  inter- 
secting each  other  in  P  revolve  round  the  fixed  points,  S,  H,  in  such 
a  manner  that  the  difference  of  the  lines  HP  and  SP  (viz.  HP — SP) 
may  be  a  constant  quantity  ;  then  the  curve  traced  out  by  the  point 
P  will  be  an  Hyperbola. 


16 


INTRODUCTION. 


In  this  case,  let  A  be  the  point  where  the  curve  cuts  SH ;  bisect 
SH  in  C,  and  take  CM=CA.  Since  CH=CS,  and  CM=CA,  HM 
will  be  equal  to  AS.  Now  when  P  comes  to  A,  HA — AS=  a  con- 
stant quantity ;  but  HA — AS=HA — HM=AM ;  v  AM  is  that  con- 
stant quantity.     Hence  AM=HP — SP. 

Let  Then 


AC  or  CM=a,  i  AN  =-CN-CA=^— a, 
SC  or  CH=6,  I  NM-ON+CM=:r+a, 
CN-ir,  j  NS  =ON— CS-^— 6, 


i 


and  PN=y;     NH  =CN+CH=^+6, 


and  HP— SP=AM= 
2AO=2a;  if  ..BF=z-\-aJ 

SP  will  be  equal  to  z — a. 


Draw  PNO  at  right  angles  to  AN,  then  we  have 

HP2=PN2+NH2,     or     (*+a)2=y2-f(:r-H>)23     (A) 

SP2=PN2+NS2,     or     {z-af^y^x-bf.     (B) 

bx 
Subtract  (B)  from  (A),  then  Aaz=ibx,  and  *— —  ;  substitute  this 

(bx      \ 2 
\-a  1  ^yi+^x+bf, 

which  reduced  is 

b*x*+2a?bx+a*=a2i/*+a9x*+2a*bx+a*b*, 
or  b*x*—a2x*— a*b*+a*=a*y\ 
i.  e.  (6*-— a2)  x  (*•—  a2)=a2y2. 


INTRODUCTION.  17 

Hence  x2 — a?  ooy2,  or  (ar— a)  x  (x+a)  ccy2 ;  i.  e.  ANxNM  ocPN2 ; 
and  since  A  lies  between  N  and  M,  the  relation  between  ANxNM, 
and  PN2,  is  the  same  with  that  in  the  Hyperbola.* 

Having  thus  established  the  identity  of  the  curves  generated  by 
these  two  different  methods,  we  now  proceed  to  demonstrate  their 
properties,  beginning  with  the  parabola. 


*  The  same  may  be  proved  geometrically,  as  follows.  The  dem- 
onstration is  applicable  either  to  the  Ellipse  or  Hyperbola. 

Take  AI=SP.  Then  IM=HP.  .-.  HP=CI+CA,  and  SP= 
CI  co  C  A. 

Now  (Euc.47.  1.)  (CI+CA)2(=HP2)=PN2-f(CN-fCS)2(=HN2); 
and,  (CI^CA)2(=SP2)=PN2+(CN^CS)2(==SN2).  That  is,  CI2+ 
2CA.CI+CA2=PN2+CN2+2CN.CS+CS2,  and  CI2— 2CA.CI+CA2 
=PN2+CN2— 2CN.CS+CS2.  Subtract,  and  4CA.CI=4CN.CS  or 
CA.CI-CN.CS,  .-.  CA  :  CN  : :  CS  :  CI,  and  CA2  ;  CN2  : :  CS2 
:CI2. 

From  A,  draw  AG  at  right  angles  to  AC ;  make  AG  a  mean 
proportional  between  AS  and  SM,  and  join  CG,  meeting  PN  in  D. 
Then  AG2=AS.SM=CS2^  CA2,  (Euc.  5.  2.  cor.)  and  CS2=CA2± 
AG2.* 

By  sim.  tri.  CA2  :  CN2  ::  CA2±AG2(CS2)  :  CN2±ND2,*  but  (as 
above)  CA2  :  CN2  : :  CS2  :  CI2,  .-.  CI2=CN2±ND2. 

In  the  first  equation  as  expanded  above,  therefore,  let  CS2±  AG2 1 
be  substituted  for  CA2,  CS.CN  for  CA.CI,  and  CN2±ND2  for  CI2, 
and  we   have    CS2±AG2+2CS.CN+CN2±ND2=PN2+CS2+2CS. 

CN+CN2,  or  ±AG2±ND2=PN2,  that  is,  AG2^ND2=PN2.  But 
(sim.  tri.)  AC2  :  AG2  ::  CN2  ^  CA2( AN.NM)  ;  AG2  *  ND2(PN2). 
But  the  ratio  AC2  :  AG2  is  constant.  Hence  AN.NMx  PN2,  which 
(N  being  between  A  and  M)  is  the  property  of  the  Ellipse,  and  (A 
being  between  N  and  M)  is  the  property  of  the  Hyperbola. 


*  The  sign  —  for  the  Ellipse,  and  +  for  the  Hyperbola. 
t  The  sign  -f  for  the  Ellipse,  and  —  for  the  Hyperbola. 
C.  S.  3 


18 


ON    THE    PARABOLA. 


CHAPTER  II. 


ON    THE    PARABOLA. 


III. 


DEFINITIONS. 


(10.)  Let  pAP  be  a  parabola  generated  by  the  lines  SP,  PM, 
moving  according  to  the  law  prescribed  in  Art.  7. ;  then  the  line 
ELF,  which  regulates  the  motion  of  the  line  PM,  is  called  the  Di- 
rectrix ;  the  point  S,  about  which  the  line  SP  revolves,  the  Focus  ; 
the  line  AZ,  which  passes  through  the  middle  of  the  curve,  the 
Axis  ;  and  the  highest  point  A,  the  Vertex  of  the  parabola. 


ON   THE    PARABOLA.  19 

(11.)  Let  fall  the  perpendicular  PN  upon  the  axis  AZ,  and 
through  the  focus  S  draw  BC  parallel  to  it,  and  meeting  the  curve 
in  the  points  B  and  C.  PN  is  then  called  the  Ordinate  to  the  axis, 
AN  the  Abscissa  ;  and  the  line  BC  is  called  the  Principal  Latus- 
rectum,  or  the  Parameter  to  the  Axis. 

(12.)  Produce  MP  in  the  direction  PW,  or,  in  other  words,  draw 
PW  parallel  to  the  axis  AZ  ;  from  any  point  Q,  of  the  parabola  draw 
QVq  parallel  to  a  tangent  at  P ;  and  through  S  draw  be  parallel  to 
Q,V.  PW  is  called  the  diameter  to  the  point  P  ;  Q,V  the  ordinate, 
PV  the  abscissa,  and  be  the  parameter,  to  the  diameter  PW. 

(13.)  Let  PT  touch  the  curve  in  P,  and  meet  the  axis  produced 
in  T,  draw  PO  at  right  angles  to  PT,  and  let  it  cut  the  axis  in  O. 
PT  is  called  the  tangent,  TN  the  subtangent,  PO  the  normal,  and 
NO  the  subnormal,  to  the  point  P. 

IV. 

On  the  Properties  of  the  Parabola. 
Property  1. 
(14.)  The  Latus-rectum  BC  is  equal  to  4AS. 

Draw  BD  (Fig.  in  page  13.)  parallel  to  LZ,  then  SB=BD=SL. 
But  since  SA=AL,  SL  is  equal  to  2AS ;  hence  SB=2AS,  and 
2SBorBC=4AS. 

This  proposition  may  be  thus  enunciated. 

The  latus-rectum  is  equal  to  four  times  the  distance  from  the  fo- 
cus to  the  vertex. 

Property  2. 
(15.)  The  tangent  PT  bisects  the  angle  MPS. 

Take  Vp  so  small  a  part  of  the  curve,  that  it  may  be  considered 
as  coinciding  with  the  tangent,  and  consequently  as  a  right  line. 
Join  Sjo,  and  draw  pm  parallel  to  AZ ;  let  fall  po,  pn,  perpendiculars 
upon  SP,  PM. 


20  ON  THE    PARABOLA. 

The  figure  Mnpm  is  a  parallelogram,  .:nM==pm  ;  and  since  po 
is  at  right  angles  to  SP,  it  may  be  considered  as  a  small  circular  arc 
described  with  radius  Sp,  .-.  So=Sp.    Also  SP=--PM,  and  Sp=pm. 

Now  Po-SP—  So=SP—  Sp,  i 
and  P«=PM— wM=SP- pm,  V  ..Vo=Vn. 
=SP— Sp,  S 

In  the  small  right-angled  triangles  Vpo,  Vpn,  we  have  therefore 
"Pp  common,  and  Po=P/i,  .-.  (47.  1.)  po=pn ;  having  .:.  their 
three  sidss  equal,  the  angle  pVo  must  be  equal  to  the  <ipVn ;  hence, 
since  pT  may  be  considered  as  the  continuation  of  the  line  Pp,  PT 
bisects  the  angle  MPS  ;  which  proposition  may  be  thus  expressed  : 

The  tangent,  at  any  point  of  the  curve,  bisects  the  angle  formed 
at  that  point,  by  the  perpendicular  to  the  directrix,  and  the  line 
drawn  to  the  focus.* 


*  The  reasoning  in  the  text,  though  perfectly  conclusive,  is  of  a 
kind  not  always  entirely  satisfactory  to  the  student,  who  is  unaccus- 
tomed to  its  use.  The  same  proposition  may  be  demonstrated  with- 
out the  use  of  indefinitely  small  arcs,  in  the  following  manner. 

It  is  first  necessary  to  establish  this  position  : — If  a  straight  line? 
not  parallel  to  the  axis  of  the  parabola,  cut  the  curve  in  one  point, 
it  will,  on  being  produced,  if  necessary,  cut  it  again. 

Let  HP,  not  parallel  to  AZ,  the  axis,  cut  the  curve  in  P.  It  will, 
on  being  produced  towards  P,  intersect  the  curve  in  some  other 
point. 

Since  HP  and  AZ  are  not  parallel,  they  will 
meet,  if  produced.  Let  them  meet  in  H. 
Draw  the  ordinate  PN,  and  take  AR  a  third 
proportional  to  AN  and  AH.  Draw  the  ordi- 
nate RQ,.  HP,  produced,  will  meet  the  curve 
in  a. 


ON   THE    PARABOLA. 


21 


E 


L I  \         *>    m 


M  F 


A       ^ 

C,Z 

s 

Eln 

N 

^4p 

0 

(16.)  Cor.  Since  the  angle  MPS  continually  increases  as  P  moves 
towards  A,  and  at  A  becomes  equal  to  two  right  angles,  the  tangent 
at  A  must  be  perpendicular  to  the  axis. 


For,  if  not,  let  it  take  some  other  direction  as  P#,  cutting  RQ,  in 

M.* 

By  Hyp.  AN  :  AH  : :  AH  :  AR  (Euc.  12.  5.)  AN  :  AH  : :  AN-f 
AH(NH)  :  AR+AH(RH)  and  AN2  :  AH2  : :  NH2  :  RH2  : :  (sim.  tri.) 
PN2  :  MR2  (Euc.  Def.  11.  5.)  AN2  :  AH2  : :  AN  :  AR  : :  (7.)  PN2  : 
QR2,  .-.  PN2  :  MR2  : :  PN2  J  QR2,  or  MR2=QR2,  which  is  impossi- 
ble, unless  HP  produced  pass  through  Q.     Therefore,  &c. 

Cor.  Hence  if  HQ,  cuts  the  curve,  AN  :  AH  : :  AH  :  AR. 

The  demonstration  is  not  essentially  dhferent  from  any  other  ar- 
rangement of  the  points  A,  P  and  Q,. 

To  prove  that  the  tangent  bisects  the  angle  SPM,  let  the  ordinate 
PN  be  drawn. 


*  M  is  taken  between  Q,  and  R,  because,  if  taken  on  the  other  side  of  &,  P#  must 
cut  the  curve. 


22 


ON   THE    PARABOLA. 


I 


Property  3. 

If  PT  meets  the  axis  produced  in  T,  then  SP=ST,  and  TN= 

2AN. 

(17.)  Since  PM  is  parallel  to  TZ,  the  angle  MPT=alternate  an- 
gle STP ;  but  (by  Prop.  2.)  <MPT=<SPT,  .-.  <STP=<SPT, 
and  consequently  SP=ST.     That  is, 

/  If  a  tangent  to  any  point  of  the  curve  cut  the  axis  produced,  the 
points  of  contact  and  intersection  will  be  equally  distant  from  the 
focus. 

(18.)  Now  (7.)  SP=AN+AS  ; 
and  ST=TA+AS. 

Hence,  since  SP=ST,  we  have  AN+AS=TA+AS,  .-.  TA=AN, 
or  TN=2AN. 


Now  if  the  tangent  does  not  bisect 
SPM,  some  other  line  which  cuts 
the  curve,  must  do  it.  Let  TP  be 
that  line,  cutting  the  curve  in  P  and 
again  in  p.*  Draw  the  ordinate 
pm. 


E 

L 

N      M 

p 

A 

'      N 
n 

p 

\ 

By  Hyp.  <SPT=<TPM=alternate  <STP. 

SP=PM(7.)=LN. 

From  ST=LN  take  AS=(7.)AL,  and   AT=AN. 


(Euc.  5.  1.)  ST= 


But  by  the 


corollary  above,  AN  :  AT  : :  AT  :  An,  or  AN=Aw,  which  is  ab- 
surd, if  TP  cuts  the  curve,  .-.  TP  is  the  tangent. 

Hence  the  tangent  bisects  <SPM. 

It  will  be  seen  that  while  we  are  demonstrating  Property  2d,  we 
at  the  same  time  prove  all  that  is  laid  down  in  Arts.  17  and  18.  In- 
deed it  would  be  better  to  demonstrate  these  latter  propositions  first, 
and  infer  Property  2d  from  them. 


*  It  is  not  essential  to  the  demonstration,  on  which  side  of  P,  p  is  taken. 


ON   THE    PARABOLA.  23 

The  subtangent  is  bisected  by  the  vertex ;  or  the  subtangent  is 
double  the  corresponding  abscissa. 

(18a.)  Hence  the  tangent  at  C,  the  extremity  of  the  latus-rectum 
meets  the  axis  in  L,  the  same  point  with  the  directrix.  For  (7.) 
SA=AL.  Hence  SL=2SA=CS,  (14.)  and  the  triangle  CSL  is 
isosceles. 

Property  4. 

(19.)  The  square  of  the  ordinate  (PN2)=  latus-rectum  x  abscissa 
(BCxAN.) 

By  Art.  7,  y2=4a#,  or  PN2=4ASxAN ;  but  by  Prop.  1,  BC= 
4AS,  .\PN2=BCxAN.  Or,  the  square  of  any  ordinate  to  the  axis 
is  equal  to  the  rectangle  of  the  corresponding  abscissa  and  the  latus- 
rectum. 


PN2  PN2 

(20.)  Cor.  Hence  BC=-^  ;  and  *  BC^-^ 

v     ;  AN  '  2AN 


Property  5. 

(21.)  The  subnormal  NO=iBC. 

Since   TPO   is   a    right   angled    triangle,    (Euclid,   8.   6.   cor.) 

PN2 

NO  :  PN  : :  PN  :  TN,  .-.NO==^  ;   but  by  Prop.   3,   TN=2AN, 

PN2 
,.NO=^— ^;  hence  (20.)  NO=iBC.     Or,  the  subnormal  is  equal 

to  half  the  latus-rectum. 

Property  6. 

(22.)  The  square  of  the  ordinate  (QV2)=4SPxPV.  (See  next 
figure.) 

Produce  Vd  to  H  ;  draw  EQ,  GV  parallel  to  PN,  and  QD  par- 
allel to  AZ ;  then  the  figures  PTHY,  PNGV  will  be  parallelograms, 
and  TH-=PV=NG ;  .-.HN+NG=HN+TH,  or  HG=TN. 


24 


ON    THE    PARABOLA. 


Let  AN=a:, 

r  HG=TN=*2AN=2# 

PV=NG=y, 

g 

HE=HG— GE  =2x—z 

dD=EG=*, 

AG=AN+NG=2:-fy 

AS=a, 

AE=AG— EG  =:r+y— z 

and.-.SP=AN+AS= 

=x-{-a. 

Now  (19.)  EQ,2=4ASxAE=4aX(3r+y— z) ;  and  by  similar  tri- 
angles,    HEd,     TNP,     we  have 

HE2  :  EQ,2  : :  TN2  :  PN2, 

i.e.  (2x — zf  :  4ax(#-fy — z)  ::    4#2  :  Aaz, 

&ax(x-\-y — z)x&x* 
Aax 


...  (2x— zj 


=4#x(#-by — z), 
or  4#2 — kxz+z^^x^+kxy — 4xs. 
Hence  z2=4#?/=Q,D2. 

T 


ON    THE    PARABOLA. 


25 


Again,  by    sim.  As,    HGY,    Q,DY,    we  have 
HG2  :  GV2      or      PN2  : :  QD2  :  DY2, 


i.  e.    4#s 


kax 


.  ■  kaxxbxy 

: :  Axy  :  DV2= — j-r-— =4ay. 


4r 


But  av2=aD2+DV2 

=4.ry+4ay=4(#-f-a)y=4SPxPV. 


That  is,  the  square  of  an  ordinate  to  any  diameter,  is  equal  to  four 
times  the  rectangle  of  the  corresponding  abscissa,  and  the  distance 
from  the  vertex  of  that  diameter  to  the  focus. 


*  This  proposition  may  be  demonstrated  geometrically  as  follows. 
In  the  first  place  QV2  ocPV. 

From  A  draw  AK,  an  ordinate  to  PV,  and  AG,  the  vertical  tan- 
gent.     Putting  L  for  the  latus-rectum,  we  have  (19)  L.AN  (or 
AT)=PN2=EF2,  and  L.AF-FQ2, 
Again,  by  sim.  tri.  PTN,  QRF 

TN(=2AN) :  FR  : :  PN(=EF)  :  FQ, 
or  AN  :  FR  : :  EF  :  2FGL 
And  the  rect's   L.AN  :  L.FR  : :  EF2 :  2EF.FQ. 


s.  c. 


26  ON    THE    PARABOLA. 

The  same  demonstration,  with  a  very  slight  alteration,  is  applica- 
ble to  the  case  when  P  and  Q,  are  on  opposite  sides  of  A. 

Property  7. 

(23.)  If  Q,V  is  produced  to  meet  the  curve  in  q,  then  qY=QV. 
(Fig.  on  p.  16.) 

Draw  qe  at  right  angles  to  AZ,  cutting  PW  in  k ;  then  He=HG-f 
Ge,  and  Ae=AG-f-Ge ;    if  therefore  Ge  or  Yk-=z.  we  have  He= 


But  L.AN=EF2   .-.L.FR=2EF.FQ. 
To  this  add  L.AT=EF2 
L.AF=Fd2 
.-.  L(TA+AF-fFR)=L.TIWEF2-f2EF.Fa+Fa2=Ea2. 

But     (sim.  tri.)AK2  :  CIV8  : :  AG2(L.AT) :  EQ2(L.TR), 

or  AK2  :  QV2  : :  AT  :  TR  : :  PK  :  PV. 
But  since    AK2  and  PK  are  constant    Q,V2  ocPV. 

Next,     QY2=4SP.PV. 

Upon  the  tangent  PT,  let  fall  the  perpendicular  SY,  from  the 
focus.  Since  STP  is  isosceles,  PT  is  bisected  by  SY.  AY  also 
bisects  PT,  since  AT=AN,  (Euc.  2.  6.)  Hence  AY  and  SY  inter- 
sect the  tangent  PT,  in  the  same  point. 

By  sim.  tri.  PN2  :  PT2(AK2)  ::  SY2(=AS.ST  Euc.  8.  6.  cor.)  : 
ST2  : :  AS  :  ST(SP). 

Hence,  because     AN=AT=PK     and    4AN=4PK, 

PN2  :  AK2  : :  4AS.AN  :  4SP.PK. 
But     PN2=4AS.AN.  .-.  AK2=4SP.PK. 
But     AK2  :  GIV2  : :  PK  :  PV  : :  4SP.PK  :  4SP.PV. 
•.  Q,V2=4SP.PV. 

These  demonstrations  are  equally  applicable  to  gY.  The  first  will 
require  one  change  of  sign,  ( — ef.fq)  but  in  all  other  respects  it  may 
remain  the  same,  only  substituting  the  small  for  the  large  letters. 
And  as  ecf  is  proved,  in  this  way,  equal  to  L.TR=EQ,2,  by  sim.  tri. 
Q,V=^V,  and  Prop.  7  requires  no  additional  demonstration. 


ON    THE    PARABOLA.  27 

2x-{-z,  and  Ae=x+ y+z.  In  this  case,  the  sign  of  z  is  changed 
throughout ;  reasoning  therefore  as  in  the  last  Property,  we  should 
have  z2  or  Vk*=4&y. 


By  sim.  As,  HGV,  Ykq,  HG2  ;  GV2  or  PN2  : :  W  :  kq\ 
i.  e.  4#2  :  4a#         : :  kxy  \  kq2. 

Hence  Ar^2= —     2      =4ay. 

But  Ytf^Vtf+kq* 

=4ry+4ay=4(#+a)y=4SPxPV. 

Since  Q,V2  and  V^2  are  each  equal  to  4SPxPV,  it  follows  that 
V<72=Q,V2,  and  consequently  V<y«Q,V.  Or,  Every  diameter  bisects 
all  lines  in  the  parabola,  drawn  parallel  to  the  tangent  at  its  vertex, 
and  terminated  both  ways  by  the  curve  ;  or  every  diameter  bisects 
its  double  ordinates. 

Property  8. 

(24.)  The  Parameter  fa  is  equal  to  4SP.     (Fig.  on  p.  16.) 

Let  be  and  PW  intersect  each  other  in  g",  tfyen  (23)  cg=gb  .% 
cg=-%bc,  and  cg*2=i&c2. 

Since  P^ST  is  a  parallelogram,  Pg-=ST=SP.  Now  (22) 
cg-2=4SPxPg-=4SPxSP=4SP2. 

Hence  £&c2=4SP2, 
and  $bc  =2SP, 
or    6c=4SP; 
that  is,  the  parameter  to  any  diameter  is  equal  to  four  times  the  dis- 
tance from  the  vertex  of  that  diameter  to  the  focus. 

(25.)  Cor.  Hence  Q,V~=(4SPxPV=)^xPV  ;  and  since  be  is 
constant  with  respect  to  the  same  diameter,  P V  ccQ,V2.  That  the 
square  of  the  ordinate  is  equal  to  the  parameter  x  abscissa,  is  there- 
fore a  general  property  of  the  Parabola. 

Property  9. 

(26.)  Draw  Q,R  parallel  to  PV,  and  meeting  PT  in  R;  then 
QRaPR2. 


28 


ON    THE    PARABOLA. 


In  this  case  (since  Q,  Vis  parallel  to  RP)  PVQJEfc  is  a  parallelogram; 
.-.PR=QV,    and    QR=PV  ;    but    (25)    PV  ocdV2,    .-.QRocPR2. 

Or,  if  diameters  be  produced  to  meet  any  tangent  to  the  Parabola, 
without  the  curve,  the  parts  of  those  diameters  between  the  curve 
and  the  tangent  will  be  as  the  squares  of  the  intercepted  parts  of  the 
tangent. 

(27.)  Cor.  From  this  it  fol- 
lows, that  if  PS,  PW,  be  two 
lines  meeting  in  a  given  angle, 
and  a  point  Q  begins  to  move 
from  P  in  such  a  manner  that 
its  distance  RQ,  from  the  line 
PS  (measured  in  a  direction 
parallel  to  PW)  shall  vary  as 
PR2,  or,  in  other  words,  that 
RQ,  TA,  SM,  &c.  shall  be 
to  each  other  as  PR2,  PT2, 
PS2,  &c.  then  the  curve  PQAM 
traced  out  by  the  motion  of 
the  point  Q,  will  be  a  Para- 
bola. 


Property  10. 

(28.)  If  Q,Y  be  a  tangent  at  Q,,  and  VP  be  produced  to  meet  it 
in  t,  then  V*  is  bisected  in  P. 

Produce  QY  to  q:  and  draw  qY  parallel  to  Yt ;  then  by  sim. 
triangles,  (since  0,^=20, V,)  QY  will  be  double  of  Q£,  and  qY 
double  of  V*. 

By  Art.  26,  VtiqY  ::  Q*2  :  QY2  : :  1  :  4 ; 
.-.P*=4?Y. 

But  Yt=$qY,  .-.  P*=4-Vif,  or  V*  is  bisected  in  P.  That  is,  if  a 
tangent  and  ordinate  to  any  diameter  be  drawn  from  the  same  point, 
their  intersections  with  the  diameter  and  diameter  produced  will  be 
equidistant  from  the  vertex  of  that  diameter. 


ON    THE    PARABOLA. 


29 


The  same  may  be  proved  of  a  tangent  at  q.  Therefore  the  tan- 
gents drawn  from  the  two  extremities  of  any  double  ordinate  inter- 
sect the  diameter  to  which  that  double  ordinate  belongs  in  the  same 
point. 


(28  a.)  This  proposition  may 
be  thus  generalized.  Let  PO  be 
any  tangent,  and  PK  any  line  in 
the  Parabola,  drawn  from  the 
point  of  contact,  and  meeting  the 
curve  in  K.  Let  AT  be  any  di- 
ameter produced  to  meet  the  tan- 
gent in  T,  and  cutting  the  line 
PK  in  I.     Then, 

AT  :  AI  : :  PI  :  IK. 


O 


/ 


K 


For  (26.)    AT  :  KO  : :  TP2  :  PO2  : :  (sim.  tri.)  IT2  :  KO2. 
Hence  (Euc.  Def.  11.  5.)  IT  is  a  mean  proportional  between  AT 


and  KO ;  or 

AT  :  IT  : :  IT  ;  KO  : :  (sim.  tri.)  PI :  PK. 

Inverted  division,  AT  :  AI  : :  PI :  IK. 

That  is,  if  from  any  point  in  the  curve,  there  be  drawn  a  tangent, 
and  also  a  line  to  meet  the  curve  in  some  other  place  ;  and  if  any 
diameter,  intercepted  by  this  line,  be  produced  to  meet  the  tangent ; 
then  will  the  curve  divide  the  diameter  in  the  same  ratio  in  which 
the  diameter  divides  the  line. 

The  same  demonstration,  very  slightly  modified,  will  apply  to  di- 
ameters intersecting  the  line  PK,  produced  either  way,  without  the 
section,  as  at,  and  dt\  of  which  it  may  be  proved  that 
at  :  ai  ::  Vi  :  i  K, 
and         at'  :  ai  : :  Pi'  :  i'K, 


Property  11. 

(29.)  Let  fall  SY  perpendicular  upon  PT,  and  let  AY  be  the  ver- 
tical tangent.     AY  and  SY  intersect  PT  in  the  same  point  Y. 


30 


ON    THE    PARABOLA. 


Since  (17.)  ST=SP,  and  SY  is  perpendicular  to  J>T,  it  will  di- 
vide the  triangle  PST  into  two  equal  triangles ;  consequently  TY= 
YP  ;  but  (18.)  TA  is  also  equal  to  AN  ;    .-.  TY  :  YP  : :  TA  : :  AN  ; 


hence,  (Euc.  6.  2.)  AY  is  parallel  to  PN,  and  consequently  perpen- 
dicular to  the  line  AZ.  That  is,  the  vertical  tangent  intersects  any 
other  tangent,  in  the  point  where  a  perpendicular  from  the  focus 
upon  that  tangent  intersects  it. 

(30.)  Cor.  Since  the  normal  PO  is  perpendicular  to  PT,  it  is 
parallel  to  SY,  .-.TS  :  SO  ::  TY  :  YP;  but  TY=YP,  .-.TS=SO. 

Hence,  (since  SP=TS=SO,)  if  a  circle  be  described  with  center 
S  at  the  distance  SP,  it  will  pass  through  the  points  P,  T,  and  O  ; 
and  the  <  OSP  at  the  center  will  be  double  of  the  angle  OTP  at 
the  circumference. 

Property  12. 
(31.)  PO  is  a  mean  proportional  between  BS  and  bg. 
Since  PY-YT,    OS=ST=SP,   and   TO=2SP=6^  (24.)    Also 
(21.)  ON=BS. 

But  (Euc.  8.  0.  Cor.)  ON  :  OP  : :  OP  :  OT 
.•.BS:OP::OP:&g-; 
that  is,  the  normal  is  a  mean  proportional  between  the  semiparame- 
ters  of  the  axis  and  the  diameter  at  the  point  of  contact. 


ON   THE    PARABOLA. 


31 


(32.)  Cor.  I.  SA,  SY  and  ST=SP,  are  severally  halves  of  ON, 
OP  and  OT.  .-.  SA  :  SY  : :  SY  :  SP  ;  and  SY2=SA.SP  ;  or  SY= 
V(SA.SP),  and  as  SA  is  constant,  SY  ocV(SP.) 

(32.a.)  Cor.  2.  Since  OP2=BS.^,  and  BS  is  constant ;  OP2  oc 
bg  cc2bg.  And  OPccV(2&g\)  That  is,  the  normal  varies  as 
the  square  root  of  the  parameter  to  the  diameter  at  the  point  of 
contact. 

(32.6.)  Cor.  3.  Since  SO=SP,  <SPO=<SOP=<OPg- ;  or,  the 
normal  bisects  the  angle  made  by  the  diameter  at  the  point  of  con- 
tact, with  the  line  drawn  from  that  point  to  the  focus. 

(32.c.)  Scholium.  In  optics,  the  angle  made  by  a  ray  of  light 
incident  upon  a  reflecting  surface,  with  a  perpendicular  to  that  sur- 
face, is  called  the  angle  of  incidence  ;  and  the  angle  made  by  a  re- 
flected ray  with  the  same  perpendicular,  is  called  the  angle  of  reflec- 
tion. It  is  a  general  law  that  the  angles  of  incidence  and  reflection 
are  equal.  Hence,  if  CAP  represents  a  concave  parabolic  mirror,  a 
ray  of  light  falling  upon  it  in  the  direction  g-P,  will  be  reflected  to  S. 
The  same  would  be  true  of  all  rays  parallel  to  gP.  Hence  the 
point  S,  in  which  all  the  rays  would  intersect  each  other,  is  called 
the  focus. 

Property  A. 

(32.d.)  Let  IH,  OR  be  any  two  diameters  intersected  by  the  par- 
allels gG,  ?Q,  in  H,  R.  Then,  IH  :  OR  : :  GH.%  :  QR.R?, 
whether  the  points  H  and  R,  be  within  or  without  the  section. 

Let  P  represent  the  parameter  to  the  diameter  P  W,  of  which  Gg 
and  Qq  are  double  ordinates. 

P        I     G     H 


32 


ON    THE    PARABOLA. 


Then  (25.)  P.PV=QV2. 
And  P.PN=ON2=VR2. 

Taking  the  diff. 

P.NV(=P.OR)=QV2  kVR2=(Euc.  5.  2.  Cor.)RQ.R?. 
In  like  manner  P.IH=GH.Hg\ 

Hence,  GH.Hg-  :  QR.R?  : :  P.IH  J  P.OR  : :  IH  :  OR. 

That  is,  the  parts  of  all  diameters,  intercepted  by  lines  parallel  to 
each  other,  whether  within  or  without  the  Parabola,  are  as  the  rect- 
angles of  the  corresponding  segments  of  the  lines. 

Property  B. 


(32.c.)  Let  the  parallels  CD,  EF  intersect  the  parallels  GH,  IK, 


in  the  points  N,  P.     Then 

CN.ND  :  HN.NG  : :  EP.PF  :  KP.PI. 
For  (32.d.)      CN.ND  :  EP.PF 
And  HN.NG  :  KP.PI 

.-.  CN.ND  :  EP.PF 


LN:OP 
LN  :  OP. 
HN.NG  :  KP.PI ; 


or  (Euc.  16.  5.)  CN.ND  :  HN.NG  : :  EP.PF  :  KP.PI ; 

Or,  the  rectangles  of  the  corresponding  segments,  into  which  par- 
allel lines  in  a  Parabola  divide  each  other,  have  to  each  other  a  con- 
stant ratio. 


ON    THE    PARABOLA. 


33 


Property  C. 

(32/.)  Let  RX  and  PW  be  the  diameters  to  which  CD  and  HG 
are  double  ordinates.  Let  P  represent  the  parameter  of  PW,  and  P' 
that  of  RX. 


Then 


CN.ND  :  HN.NG  : :  F  ;  P. 
p 


By  reasoning  like  that  employed  in  Prop.  A,  it  may  be  shown 
that 

CN.ND=P'.IN  and  HN.NG=P.IN, 

.-.  CN.ND  :  HN.NG  : :  P'.IN  :  P.IN  : :  F  :  P. 


This  proposition  may  be  thus  enunciated  : 

If  any  two  straight  lines,  which  meet  the  curve  in  two  points,  in- 
tersect each  other,  the  rectangles  of  their  corresponding  segments 
will  be  as  the  parameters  of  the  diameters,  to  which  those  lines  are 
double  ordinates. 

The  last  two  propositions,  like  Property  A,  are  applicable  to  lines 
both  within  and  without  the  section,  and  the  diagrams  are  lettered 
in  such  a  manner  that  the  demonstration  may  apply  to  either  case. 
.    C.S.  5 


34 


ON    THE    ELLIPSE. 


CHAPTER  III. 
ON   THE   ELLIPSE. 


V. 


DEFINITIONS. 

(33.)  Let  APMO  be  an  Ellipse  generated  by  the  revolution  of 
the  lines  SP,  HP,  about  the  fixed  points  S,  H,  according  to  the  law- 
prescribed  in  Art.  8. ;  then  B  Q, 
the  line  AM,  which  passes 
through  the  two  foci  S  and 
H,  is  called  the  Axis  Major ; 
and  if  through  the  center  C 
a  line  BCO  be  drawn  at 
right  angles  to  AM,  it  is 
called  the  Axis  Minor  of 
the  Ellipse. 


(34.)  From  any  point  P  let  fall  the  perpendicular  PN  upon  the 
axis  major  AM,  and  through  the  focus  S  draw  the  straight  line  LST 
parallel  to  it.  PN  is  then  called  the  ordinate  to  the  axis  ;  AN,  NM, 
the  Abscissas  ;  and  the  line  LST  is  called  the  latus-rectum,  or  the 
Parameter  to  the  Axis. 


(35.)  Draw  any  line  PCG  through  the  center,  and  another  line 
DCK  parallel  to  a  tangent  at  P ;  draw  also  Qv  parallel  to  DCK. 
PCG  is  then  called  a  Diameter,  and  DCK  the  Conjugate  diameter 
to  PCG ;  dv  is  called  an  Ordinate  to  the  diameter  PCG,  and  Pv, 
vG,  the  Abscissas. 


ON   THE    ELLIPSE. 


35 


YI. 

On  the  Properties  of  the  Ellipse. 
Property  1. 

(36.)  If  SB,  HB,  are  drawn  from  the  foci  to  the  extremity  of  the 
axis  minor,  then  SB,  HB,  are  each  equal  to  AC. 

Since  SC=CH,  and  BC  is  common  to  the  two  right-angled  tri- 
angles BCS,  BCH,  SB  must  be  equal  to  BH  ;  .-.  SB+BH-2SB  or 
2BH. 

Again,  by  Sect.  2.  Art.  8.  SP+PH=AM=2AC  ;   and  when  P 

B 


- 

t 

~^\P 

S 

c 

H           1 

T 

( 

> 

M 


comes  to  B,  SB+BH=2AC ;  hence  2SB  or  2BH=2AC ;  .-.  SB  or 
BH=AC. 

That  is,  the  distance  from  either  focus  to  the  extremity  of  the 
axis  minor  is  equal  to  the  semi-axis  major. 

Property  2. 

(37.)  MSxSA=BC2 

For  BC2=SB2— SC2  (Euc.  47.  1.) 

=AC2— SO2  (by  Prop.  1.) 

=(AC+SC)X(AC— SO) 

=(CM+SC)x(AC— SC)  (forCM=AC) 

=MSxSA. 

That  is,  the  rectangle  of  the  focal  distances  from  the  vertices  is 
equal  to  the  square  of  the  semi-axis  minor. 


36  ON   THE    ELLIPSE. 

(38.)  Cor.  In  the  same  manner,  it  might  be  shown,  that  AH  x 
HM=BC2. 

Property  3. 

(39.)  The  latus-rectum  LST  is  a  third  proportional  to  the  major 
and  minor  axes. 


For 
SL+LH=2AC(by  const: 

.-.LH=2AC  —  SL, 
and  LH2=4AC2— 4AC 

xSL+SL2, 


Again, 
LH2=SL2+SH2  (Euc.  47.  1.) 
=SL2+4SC2  (for  SH=2SC) 
=SL2+4(SB2— BC2) 
=SL2+4(CA2— BC2). 


Hence  4AC2— 4ACxSL+SL2=SL2+4AC2— 4BC2 ; 
.-.4ACxSL=4BC2. 

And  putting  this  equation  into  a  ;  2AC  :  2BC  : :  2BC  :  2SL, 

proportion,  we  have  \      or  AM  :  BO    : :  BO    :  LT. 

Therefore  the  latus-rectum  is  a  third  proportional  to  the  major  and 
minor  axes. 

(39a.)  AS.SL=iLT.AM. 
For  (39)   B02=LT.AM. 

.-.  iB02(=BC2)=(37)  AS.SM=^LT. AM. 

Property  4. 

(40.)  Produce  SP  to  p  ;  then  if  YZ  bisects  the  angle  HP/?,  it  will 
be  a  tangent  to  the  Ellipse  in  P.     (Fig.  in  page  29.) 

For  if  YZ  does  not  touch  the  ellipse,  let  it  cut  it  in  Q, ;  take  Pp= 
PH,  and  join  pB,  QS,  QH,  and  Qp.  Since  P^=PH,  PZ  com- 
mon, and  <CpPZ=HPZ,  the  side  pZ  will  be  equal  to  ZH  ;  and  the 
<s  PZjo,  PZH,  will  be  equal  and  consequently  right  <s.  Again, 
since  pZ=ZH,  ZQ,  common,  and  <s  QZp,  Q,ZH  right  <s,  the  side 
Qp  is  equal  to  the  side  Q,H. 

Now  (by  Euc.  20.  1.)  SQ,+Qp  is  greater  than  Sp  or  SP-fPp 
or  SP+PH ;  but  QH=Qp  ;  therefore  SGl+aH  is  greater  than 
SP+PH  ;  but  if  Q,  is  a  point  in  the  curve,  SQ,-J-Q,H  must  be  equal 


ON   THE    ELLIPSE. 


37 


to  SP+PH  ;  Q,  therefore  is  not  a  point  in  the  curve.  In  the  same 
manner  it  might  be  proved  that  YZ  does  not  meet  the  curve  in  any 
other  point  on  either  side  of  P,  it  must  therefore  be  a  tangent  at  P. 

Hence,  if  from  the  foci  two  straight  lines  be  drawn  to  any  point 
in  the  curve,  the  straight  line  bisecting  the  angle  adjacent  to  that 
contained  by  these  lines,  is  a  tangent. 

(41.)  Cor.  1.  It  follows,  from  the  above,  that  the  <SPY=<HPZ  ; 
for  <SPY=vertical  <pPZ  ;  but  pPZ=HPZ  ;  .-.  SPY=HPZ  ;  and 
this  is  a  distinguishing  property  of  the  ellipse  ;  viz.  That  lines 
drawn  from  the  foci  to  any  point  in  the  curve  make  equal  angles 
with  the  tangent  at  that  point. 

(41.a.)  Hence,  also,  (see  Art.  32.c.)  if  rays  of  light  proceed  from 
one  focus  of  a  concave  ellipsoidal  mirror,  they  will  be  reflected  by 
the  mirror  into  the  other  focus. 

(42.)  Cor.  2.  When  P  comes  to  A  or  M,  the  angle  HPp  becomes 
equal  to  two  right  angles  ;  at  A  or  M,  therefore,  the  tangent  is  per- 
pendicular to  the  axis  AM. 

Property  5. 


(43.)  If  tangents  be  drawn  at  the  extremities  of  any  diameter  of 
an  Ellipse,  they  will  be  parallel  to  each  other. 


38  ON   THE    ELLIPSE. 

Complete  the  parallelogram  SPHG,  of  which  SP,  PH  are  two 
sides,  and  join  PG  ;  then  since  the  opposite  sides  of  parallelograms 
are  equal  to  each  other,  SG+GH  is  equal  to  SP+PH,  and  conse- 
quently G  is  a  point  in  the  Ellipse ;  and  since  the  diagonals  of 
parallelograms  bisect  each  other,  SH  is  bisected  in  C  ;  therefore  C 
is  the  center  of  the  Ellipse,  and  PG  a  diameter  (35). 

Now  let  the  tangents  ef,  gh  be  drawn  at  the  extremities  of  the  di- 
ameter PG;  then  by  Art.  41.  the  <SPe=<HP/;  but  SPe+HP/ 
is  the  supplement  of  <SPH ;  .-.  SPe=4  supplement  of  SPH.    For 


the  same  reason,  the  <HGA=£  supplement  of  <SGH ;  but  the  <s 
SGH,  SPH  are  equal,  being  opposite  <s  of  a  parallelogram ;  hence 
the  <SPe=<HGA.  Again,  since  SP  is  parallel  to  GH,  the 
<SPG=<PGH  ;  therefore  SPe+SPG=HGA+PGH,  or  GPe=PGA, 

and  consequently  ef  is  parallel  to  gh.     Therefore,  if  tangents,  &c. 

(44.)  Hence,  if  tangents  be  drawn  at  the  extremities  of  any  two 
diameters  of  an  Ellipse,  they  will  form  a  parallelogram  (eghf). 


ON   THE    ELLIPSE. 


Property  6. 


(45.)  If  SP  intersects  the  semi-conjugate  diameter  (CD)  in  E, 
then  PE  is  equal  to  the  semi-major  axis  (AC). 

Draw  HI  parallel  to  CD  or  ef,  then  the  <PIH=altemate  <SPe, 
and  <PHI=alternate<HP/;  but  <SPe=<HP/,  .-.  <PIH=<PHI, 
and  consequently  P1=PH.  Again,  since  CE  is  parallel  to  HI,  and 
SC=CH,  SE  must  be  equal  to  EI. 

Hence  FuBf* 
EI=SI. 

,(PI+EI)orPE=Si±^( 

SP+PH     2AC 
______  AC. 

Therefore,  if  from  the  extremity  of  any  diameter,  a  line  be  drawn  to 
the  focus,  meeting  the  conjugate  diameter,  the  part  intercepted  by 
the  conjugate  will  be  equal  to  the  semi-major  axis. 

Property  7.  .  1 

(46.)  If  the  ordinate  PN  be  drawn  to  the  major  axis,  then  ANx 
NM  :  PN2  : :  AC2  :  BC2  : :  AM2  :  BO2.     (Fig.  on  p.  32.) 


Let 
AC  or  CM=a, 
SC  or  CH=6, 

CN=*, 
and  PN=2/; 


Then,  by  Art.  8,  page  7, 
(a2— 62)  X  (a— x)  X  (a+ #)=a2y2 , 
or  (AC2— SC2)  x  AN  X  NM=AC2  x  PN2. 
But  AC2— SC2=SB2— SC2=BC2 ; 


.-.  BC2xANxNM=AC2xPN2, 
and  ANxNM  :  PN2  : :  AC2  :  BC2  : :  AM2 ;  BOa. 

That  is,  as  the  square  of  the  axis  major  is  to  the  square  of  the 
axis  minor,  so  are  the  rectangles  of  the  abscissas  of  the  former,  to 
the  squares  of  their  ordinates. 


40 


(47.)  Cor.  Since  ANxNM=(AC— GN)x(AC+CN)=AC2— CN2 ; 
.-.  AC2— CN2  :  PN2  : :  AC2  :  BC2. 

Property  A. 

(47.a.)  If  from  P,  the  line  PR=AC,  be  drawn  to  BO,  then  PI= 
BC. 

For  (sim.  tri.)     Rrc2(PR2  — Pn2)  :  PN2  : :  PR2  :  PP. 
That  is,    AC2— CN2  ;  PN2  : :  AC2  :  PI2. 
But  (47)  AC2— CN2  :  PN2  : :  AC2  :  BC2. 
.-.PI2=BC2     and    PI=BC. 

Or,  if  from  any  point  in  the  Ellipse,  a  line  be  drawn  to  the  minor 
axis,  equal  to  the  semi-major,  the  part  intercepted  between  that  point 
and  the  major  is  equal  to  the  semi-minor  axis. 

Property  8. 

(48.)  If  the  ordinate  ¥?i  be  drawn  to  the  minor  axis,  then  Buy, 
nO  :  Prc2  : :  BC2  :  AC2. 

In  this  case,  P?i=CN,  and  Crc=PN  ;  .-.  AN  X  NM=AC2-~ Prc2, 
(47) ;  hence,  by  substitution  in  Art.  47,  we  have, 

AC2— Pn2:  Crc2::         AC2         :  BC2, 

...  AC2  :  AC2— Pn2  : :        BC2        :  C/i2 ; 
andPrc2:  AC2::    BC2— Cw8   :  BC2,* 


*  For  AC2 :  AC2 -(AC2— Pw2)  or  Prc2  : :  BC8 ;  BC2— Cn8,  .-.  (in- 
vertendo)  Prc2 :  AC2  : :  BC2— Cn2  :  BC2. 


ON    THE    ELLIPSE. 


41 


Hence    BnxnO  :  Prc2  : 


(BC— Cro)x(BC+Cn)  :  BCS 


BnxnO 
BC2  :  AC2. 


:BC2, 


That  is,  as  the  square  of  the  axis  minor  is  to  the  square  of  the 
axis  major,  so  are  the  rectangles  of  the  abscissas  of  the  former,  to 
the  squares  of  their  ordinates. 

(49.)  Cor.  Since  B/ixnO=BC2— Crc2,  we  have 
BC2— Cn2  :  Pn2  : :  BC2  :  AC2. 


Property  9. 

(50.)  Describe  the  circle  ARML  upon  the  major  axis  AM,  and 
draw  an  ordinate  Q,PN  cutting  the  ellipse  in  P  ;  then  Q,N  ;  PN  : : 
AC  :  BC. 


By  Art.  46.     ANxNM  :  PN2  : :  AC2 :  BC2. 
But  by  Prop,  of  circle,  AN  x  NM=aN2 ; 

.-.  QN2  :  PN2  : :  AC2  :  BC3, 
and  GIN   :  PN   : :  AC   :  BC. 

In  like    manner,   it  may  be  shown  that    qn  :  Pn  : :  BC  :  AC. 
Hence,  if  a  circle  be  described  on  either  axis,  then  any  ordinate  in 
c.  s.  6 


42 


ON   THE    ELLIPSE. 


the  circle,  is  to  the  corresponding  ordinate  in  the  Ellipse,  as  the  axis 
of  that  ordinate  is  to  the  other  axis. 

(51.)  Cor.  1.  Since  RC  is  equal  to  AC,  QN  :  PN  : :  RC  :  BC. 
Hence  it  appears,  that  if  upon  AM  as  diameter,  a  circle  be  described, 
and  if  B  be  a  given  point  in  the  line  RC  ;  then  if  the  ordinates  of 
this  circle  are  diminished  in  the  given  ratio  of  RC  :  BC,  the  curve 
APBM  passing  through  the  extremities  of  these  lesser  ordinates, 
will  be  an  ellipse,  whose  axis  major  is  to  the  axis  minor  in  the  same 
given  ratio. 

Also,  if  upon  BO  a  circle  be  described,  and  if  A  be  a  point  in  Cr, 
produced ;  then  if  the  ordinates  of  this  circle  are  increased  in  the 
given  ratio  of  Cr  :  CA,  the  curve  BPAO,  passing  through  the 
extremities  of  these  greater  ordinates  will  be  an  ellipse,  whose  axis 
minor  is  to  its  axis  major  in  the  same  given  ratio. 

(52.)  Cor.  2.  From  hence  also  it  may  be  shown,  that  the  ortho- 
graphic projection  of  a  circle  upon  a  plane  will  be  an  ellipse.  Sup- 
pose the  circular  plane  ARML  to  be  inclined  to  the  plane  of  this 


paper  in  such  a  manner,  that  the  semicircle  ARM  may  be  above 
the  paper,  and  the  semicircle  ALM  below  it,  and  let  AM  be  the  com- 
mon intersection  of  the  two  planes.    Let  the  semicircle  ARM  be 


ON   THE    ELLIPSE.  43 

projected  downwards  upon  the  plane  of  the  paper,  by  drawing  perpen- 
diculars Q,P,  RB,  from  each  point  of  the  circle,  and  let  the  semicir- 
cle ALM  be  projected  upwards,  by  drawing  the  perpendiculars  qp, 
LO,  &c. ;  then  the  curve  ABMO,  marked  out  by  this  projection, 
will  be  an  ellipse.  For  draw  Q,N,  RC,  at  right  angles  to  AM,  and 
join  PN,  BC  ;  then  the  angles  Q.NP,  RCB,  will  measure  the  in- 
clination of  the  planes,  and  PN,  BC  will  be  perpendicular  to  their 
common  intersection  AM.  Now  Q,N  :  PN  : :  rad.  ;  cos.  <Q,NP,  and 
RC  :  BC::rad.  :  cos.  RCB(=QNP) ;  .-.  QN  :  PN  : :  RC  or  AC  :  BC  ; 
and  consequently,  the  four  lines  QN,  PN,  RC,  BC,  bear  the  same 
relation  to  each  other  and  to  AM  as  they  did  in  Cor.  1  ;  hence 
P,  B,  &c.  are  points  in  an  ellipse.  In  the  same  manner  it  may  be 
proved,  that  the  semicircle  ALM  is  projected  into  a  semi-ellipse 
AOM  ;  and  thus  the  whole  circle  ARML  is  projected  into  an  ellipse 
ABMO,  whose  axis  major  is  AM. 

This  proposition  is  likewise  manifestly  true,  when  the  plane  of 
projection  does  not  cut  the  circle,  or  cuts  it  unequally. 

Property  10. 

t 

(53.)  Let  PCG  be  any  diameter  of  an  ellipse,  and  DCK  its  conju- 
gate diameter  ;  draw  the  ordinate  Q,V,  then 

PG2  :  DK2  : :  PV.VG  :  QV2. 

Let  the  circle  AqMg  be  projected  into  the  ellipse  AQMG,  accord- 
ing to  the  principles  just  now  laid  down,  and  let  the  diameter  pCg 
of  the  circle  be  projected  into  the  diameter  PCG  of  the  ellipse. 
Draw  the  diameter  dCk}  at  right  angles  to  pVg,  and  qv  parallel  to 
dCk,  and  let  dCk,  qv  be  projected  into  DCK,  QV ;  then  since  parallel 
lines  are  projected  into  parallel  lines,  QV  will  be  parallel  to  DCK. 
Now  it  is  evident  that  a  tangent  to  the  circle  at  p  would  be  projected 
into  a  tangent  to  the  ellipse  at  P ;  dCk  and  qv  therefore  being  par- 
allel to  a  tangent  at  p,  (for  they  are  both  perpendicular  to  pCg) 
DCK  and  QV  will  both  be  parallel  to  a  tangent  at  P  ;  hence  DCK  is 
the  conjugate  diameter,  and  QV  the  ordinate,  to  the  diameter  PCG. 
Again,  since  Qq  is  parallel  to  dD  (for  they  are  both  at  right  <s  to 


44 


the  plane  of  the  ellipse,)  and  QT  parallel  to  DC,  the  plane  ©f  the 
triangle  dDC  must  be  parallel  to  the  plane  of  the  figure  QqvY  ;  but 
qv  is  parallel  to  dO  ;  if  therefore  Q,V  and  qv  are  produced  till  they 
meet  in  L,  they  will  form  a  triangle  Qdh,  similar  to  the  triangle 
cZDC ;  and  since  qL  is  in  the  plane  of  the  circle,  and  Q,L  in  the 
plane  of  the  ellipse,  the  point  L  must  be  in  the  common  intersection 
(AM  produced)  of  those  planes.  Now  pP,  vY  being  perpendicular 
to  the  plane  of  the  ellipse,  are  parallel  to  each  other,  and  to  the  lines 
Qq,  dD  ;  hence  it  appears  that  the  triangles  Q.gL,  YvL,  dDC,  And 
the  triangles  /?PC,  vYC,  are  respectively  similar  ;  we  have  then,  by 
property  of  circle, 


Cp2  :  Cp2— Cv*(pv.vg  Euc.  5.  2.  Cor.)  : :  Cd2  :  qv*. 

CV2(PV.VG) 


But,  (sim.  tri.)  Cp*  :  C^2— Cv2  : :  CP2  :  CP 
And  CcZ2  :  qv*  : :  CD*  :  dV2. 

.-.  CP2  :  PV.YG  : :  CD2  •  aV2 ; 
Or,  CP2  ;  CD2  : :  PG2  :  DK2  : :  PY.VG 


avs 


The  same  demonstration  is,  obviously,  applicable  to  Ym,  and  CK. 
Consequently,  as  the  square  of  any  diameter  is  to  the  square  of  its 
conjugate,  so  are  the  rectangles  of  its  abscissas  to  the  squares  of  their 
ordinates. 


ON   THE    ELLIPSE.  45 

(54.)  Cor.  Since  any  diameter  in  the  Ellipse  is  the  projection  of 
a  corresponding  diameter  in  the  circle,  and  since  all  the  diameters 
of  the  circle  are  bisected  in  the  center  ;  it  follows  that  all  diameters 
of  the  Ellipse  are  bisected  in  the  center.  For  similar  reasons,  every 
diameter  in  the  ellipse  bisects  its  double  ordinates,  or  lines  drawn  in 
the  Ellipse,  parallel  to  the  tangent  at  its  vertex. 

Property  B. 

(54.a.)  Let  PG,  DK  be  any  two  conjugate  diameters,  and  EF, 
aS  any  lines  parallel  to  PG,  DK,  intersecting  each  other  in  M. 
Then  PG2  :  DK2  : :  EM.MF  :  QM.MS. 

Draw  the  ordinate  EN. 


Then  (53.)  PG2  :  DK2  : :  PN.NG(CP2— CN2)  :  EN2. 

Also  PG2  :  DK2  : :  CP2— CV2  :  QV2. 

.-.  (Euc.  19.  5.)  PG2  :  DK2  : :  CN2— CV2(=Ew2— Mn2) 

:  QV2— EN2(=MV2). 
That  is,  (Euc.  5,  2.  Cor.)  PG2  :  DK2  : :  EM.MF  :  QM.MS. 

The  same  demonstration  is  applicable,  (with  a  single  change 
of  sign)  to  E'F',  which  intersects  QV  in  M'  without  the  Ellipse. 
Wherefore,  if  straight  lines  in  the  Ellipse  parallel  to  two  conjugate 
diameters  intersect  each  other,  either  within  or  without  the  Ellipse, 
the  rectangles  of  their  corresponding  segments  are  to  each  other  as 
the  squares  of  the  diameters  to  which  they  are  parallel. 

(54.6.)  Cor.  PG2 :  DK2  or  PC2 :  DC2 : :  En2~Mn2 :  Q,V2— MV2. 


46 


ON   THE    ELLIPSE. 


Property  11. 

(55.)  If  Q,T,  PT.  are  tangents  to  the  circle  and  ellipse  in  the  points 
Q,  and  P,  they  will  meet  in  the  axis  produced  at  T  ;  and  CA  will  be 
a  mean  proportional  between  CN  and  CT.  And  if  Kt,  let,  are  tan- 
gents to  the  points  K  and  k,  BC  will  be  a  mean  proportional  between 
Cn  and  Ct. 

Let  Q,T  be  a  tangent  to  the  circle  in  Q,  and  join  TP.  If  TP 
does  not  touch  the  ellipse,  let  it  cut  it  in  P,  p  ;  and  through  p  draw 
the  ordinate  mpqr,  meeting  TQ,  produced  in  r. 


QLy^ 

t 
r       R 

0 

B         Y^\ 

i/Cdf* 

i 

P 

T  *r                 \ 

^^y\ 

A  I             B 

*       1 

n 

C                          j 

M 


By  sim.  As,  TNP,  Tmp  ;  TNQ,  Tmr ;  we  have 
TN  :  Tm  i :  PN  :  pro, 
and 

TN:Tro::CiN:rro; 

.-.  PN  :pm::  QN  :  rw=^pX^-  ;   but  by  Art.  50,  the  ordi- 

nates  of  the  circle  and  ellipse  are  to  each  other  in  a  given  ratio  ; 

therefore  GIN  :  PN  : :  qm  :  pm,  or  qm=— -=    — .    Hence  rm=qm, 

which  is  impossible  ;  .-.  TP  does  not  cu*  the  ellipse,  and  consequently 


ON    THE    ELLIPSE. 


47 


is  a  tangent  to  it  in  P  ;  and  since  TQ,  touches  the  circle  in  Q,,  CQ/T 
is  a  right-angled  triangle,  .-.  (Euc.  6.  8.)  CN  :  CQ, : :  Ca  :  CT  ; 
but  CQ=CA,  hence  CN  :  CA  : :  CA  :  CT,  and  CNxCT=CA2. 

(56.)  Upon  the  axis  minor  BO  describe  the  circle  BkO  ;  draw 
the  ordinate  Kkn,  and  join  Ck.  By  Art.  48,  we  have  BnxnO  : 
K?i2  : :  BC2  :  AC2 ;  by  the  same  process,  therefore,  as  that  in  Art. 
50,  it  may  be  proved  that  Kn  is  to  kn  in  a  given  ratio.  Draw  Kt, 
kt,  to  the  ellipse  and  the  circle  ;  then  proceeding  in  the  same  man- 
ner as  in  the  former  part  of  this  demonstration,  we  might  show  that 
they  will  meet  in  the  minor  axis  produced.  Since  kt  touches  the 
circle,  Ckt  is  a  right-angled  triangle,  .-.  Cn  :  Ck  ::  Ck  l  Ct]  but 
CA;=BC,  .-.  Cn  :  BC  : :  BC  :  Ct,  or  Cn  xC*=BC2. 

Hence,  if  a  tangent  and  an  ordinate  to  either  of  the  axes  be  drawn 
to  any  point  of  the  Ellipse,  meeting  that  axis  and  axis  produced, 
then  the  semi-axis  is  a  mean  proportional  between  the  distances  of 
the  two  intersections  from  the  center. 

(57.)  Cor.  In  the  right-angled  triangle  CQT,  (Euc.  8.  6.)  CN  ; 
QN  : :  Q,N  :  NT,  .-.  CN  X  NT=Q,N2=CQ,2— CN2=AC2— CN2.  For 
the  same  reason,  Cn  X  nt=BC2 — Cn2. 

Property  C. 


(57. a.)  Let  TLG  be  the  focal  tangent,  or  the  tangent  drawn  at 

F 


48  OK  THE    ELLIPSE. 

the  extremity  of  SL,  the  ordinate  from  the  focus.     Let  NPG  be  any 
ordinate  produced  to  meet  the  tangent  TLG.     Then  SP=NG. 

If  AI  be  taken  equal  to  SP,  IM=HP. 

.-.  SP-CA+CI,  and  HP=CA— CI. 

(Euc.  47.  1.)  (SP2)(CA+CI)2=PN2+(CS+CN)2(NS2). 

And  (HP2)  (CA— CI)2=PN2+(CS— CN)8(HN2). 

Expand  and  subtract  4CA.CL=4CS.CN,  and  CA.CI=CS.CN. 

.-.  ON  :  CI(SP— CA)  : :  CA  :  CS  : :  CT  :  CA  (55.) 

.-.  CN+CT(TN) :  SP  : :  CT  :  CA. 

Again,  (55.)  CS.CT=AC2,  and  CS.ST-CA2— CS2=BCS  (3G,  and 
Euc.  47.  1.)  .-.  CS.CT  : :  CS.ST  or  CT  :  ST  : :  AC2 :  BC2  : :  AC  : 
SL(39.) 

.-.  ST  :  SL  : :  CT  :  AC  : :  TN  :  SP. 

But  (sim.  tri.)  ST  :  SL  : :  TN  :  NO. 
.-.  SP=NG ; 

That  is,  the  distance  from  the  focus  to  any  point  of  the  curve  is 
equal  to  the  ordinate  to  that  point,  produced  to  meet  the  focal  tan- 
gent. 

Cor.  1.  AS-AE,  SM=MF,  and  C*=AC. 

Cor.  2.  Hence,  also,  since  TN  :  NG  is  a  constant  ratio,  TN  : 
SP  is  a  constant  ratio.  Therefore,  if  a  line  be  drawn  through  T  per- 
pendicular to  AC  produced,  the  distance  of  the  point  P  from  this 
line  (=TN)  is  in  a  constant  ratio  to  SP,  the  distance  of  the  same 
point  from  the  focus.  This  ratio,  being  (by  demonstration  above) 
=CT  :  CA,  is  a  ratio  of  greater  inequality.  This  perpendicular  is 
the  directrix  of  the  Ellipse.     (See  Art.  138,  et  seq.) 

Property  12. 

(58.)  If  PCG,  DCK,  be  conjugate  diameters  of  the  ellipse,  and 
PF  perpendicular  to  CK,  then  POxPF=BC2. 


ON    THE    ELLIPSE. 


49 


Draw  Cy  parallel  to  PF.     Then  because  PO  is  parallel  to  Cy, 

and  Ct  parallel  to  PN,  and  the  <s  Cyt,  PNO  right  <s,  the  triangles 

PON,  are  similar  ;    .-.  Ct :  Cy  : :  PO  :  PN  ;   but  Cy=PF,  and 

PN=Cw,  (being  opposite  sides  of  parallelograms) ;    ,%Ct :  PF  :: 


PO  :  Cn,  or  PFxPO=Cwx07=-BC2  by  Art.  56.  That  is,  if  from 
the  extremity  of  any  diameter,  a  perpendicular  be  drawn  to  its  con- 
jugate ;  then  the  rectangle  of  that  perpendicular  and  the  part  of  it 
intercepted  by  the  axis  major,  will  be  equal  to  the  square  of  the 
semi-axis  minor. 


Property  13. 

(59.)   Draw  the  ordinates  DL,  PN,  to  the  major  axis,  then 
+CL2=AC2,  and  PN2+DL2=BC2. 
By  Art.  47, 

AC2— CL2     :  DL2  : :  AC2  :  BC2,  (A) 

AC2— CL2 :      DL2      : :  CN  X  NT  :  PN2, 


and  AC2— CN2 

.-.  AC2— CL2 : 

or  AC2— CL2:CNxNT 


DL2 
CL2 


:  PN2, 

:  NT2    by  sim.  As  ) 
DCL,  PTN.  $ 


c.s. 


NT 

NT 

:CN, 

CT 

:CN, 

CTxCN 

:  CN2, 

AC2 

:CN2 

(55.) 

50  ON   THE    ELLIPSE. 

AP2    ri2    CNxNTxCL*    CNxCL» 
Hence  AC2— CL2= NT*- 

From  which  we  have, 

CL2 :  AC2— CL2 
and  compdo,  AC2  :  AC2— CL2 


Hence  AC2— CL2=CN2,  or  CN2+CL2-AC2. 

(60.)  Since  AC2 — CL2=CN2,  by  substitution  in  proportion  (A) 
we  have  CN2  :  DL2  : :  AC2  :  BC2 ;  but  AC2— CN2  :  PN2  : :  AC2  : 
BC2,  .-.  CN2  :  DL2  : :  AC2— CN2  :  PN2.     And  alternately, 

CN2  :  AC2— CN2  : :       DL8         :  PN2, 
Compdo,   CA2 :  AC2— CN2 : :  DL2+PN2  :  PN2  ; 

.-.  AC2  :  DL2+PN2  : :  AC2— CN2 :  PN2  : :  AC2 :  BC2. 
Hence  DL2+PN2=BC2. 

Hence  if  ordinates  to  either  axis  be  drawn  from  the  extremities  of 
any  two  conjugate  diameters,  the  sum  of  their  squares  will  be  equal 
to  the  square  of  half  the  other  axis. 

Property  14. 

(61.)  PC2+CD2=AC2+BC2,  PG  and  DK  being  conjugate  diam- 
eters.    (See  last  Fig.) 

For  by  Prop.  13.   CN2+CL2=AC2, 
and  PN2+DL2=BC2  ; 
.-.  CN2+PN2+CL2+DL2=AC2-f  BC2, 
or   CP2+CD2=AC2+BC2. 

Therefore  the  sum  of  the  squares  of  any  two  semi-conjugate  di- 
ameters is  equal  to  the  sum  of  the  squares  of  the  semi-axes. 

Property  15. 
(62.)  CD  x  PF«AC  x  BC.    (See  last  Fig.) 


ON 

THE    ELLIPSE. 

In  case  2.  of  Prop.  13.  it  was  proved 

that 

CN2 :  DL2  : : 

AC2  :  BC2 ; 

.-.CN   :DL   :: 

AC  .:  BC, 

andCN:AC    :: 

DL  :  BC. 

By  similar  As,  TCy,  DCL,   CT  :  Cy(PF) 

::CD 

Hence  we  have,    CN  : 

AC      : 

:DL: 

BC, 

and    CT: 

PF      : 

:CD: 

DL, 

.-.  CNxCT(AC2)  : 

ACxPF    : 

:CD: 

BC, 

or    AC: 

PF       : 

:CD: 

BC. 

.-.  CDxPF==ACxBC. 

51 


DL. 


That  is,  if  from  the  extremity  of  any  diameter,  a  perpendicular 
be  drawn  to  its  conjugate,  the  rectangle  of  that  perpendicular  and 
the  semi-conjugate,  is  equal  to  the  rectangle  of  the  semi-axes. 

(63.)  Cor.  From  this  it  appears,  that  all  the  parallelograms  cir- 
cumscribing the  ellipse,  and  having  their  sides  drawn  through  the  ex- 
tremities of  any  diameter  and  its  conjugate,  are  equal  to  each  other 
and  to  the  parallelogram  described  about  the  major  and  minor  axes. 
For  the  parallelogram  eghf  described  about  the  conjugate  diameters 
PCG,  DCK,  is  equal  to  four  times  eDCP=4CDxPF=4ACxBC= 
right-angled  parallelogram  whose  sides  are  2AC  and  2CB=  paral- 
lelogram described  about  the  major  and  minor  axes. 

Property  16. 

(64.)  If  SY,  HZ,  be  drawn  from  the  foci  perpendicular  upon  the 
tangent  YZ,  the  points  Y,  Z,  will  be  in  a  circle  described  upon  the 
major  axis  AM.     (Fig.  in  p.  52.) 

Join  YC,  produce  HP  to  W,  making  PW=PS,  and  join  WY. 

By  Prop.  4.  PY  bisects  the  <  SPW ;  and  since  SP=PW,  and 
PY  is  common,  WY  will  be  equal  to  YS,  and  <  WYP=<SYP=  a 
right  angle  ;  hence  WYS  is  a  straight  line.     Now  since  WY=YS, 


52 


ON   THE    ELLIPSE. 


and  SC=CH,  CY  must  be  parallel  to  WH,  and  .-.  SC  :  SH  : :  CY  : 
HW,  but  SC=4SH  ;  .-.  CY=£HW=4(HP+PS)=4AM=AC  ;  hence 
Y  is  a  point  in  the  circle  whose  center  is  C  and  radius  CA.  In 
the  same  manner  it  may  be  proved  that  Z  is  a  point  in  the  same 
circle. 


Therefore,  if  perpendiculars  be  dropped  from  the  foci  upon  any- 
tangent  to  the  Ellipse,  the  intersections  of  those  perpendiculars  with 
the  tangent  will  be  in  the  circumference  of  a  circle  described  upon 
the  axis  major. 


Property  17. 


(65.)  SYxHZ=BC2 


Since  the  <HZP  is  a  right  angle,  it  must  be  in  a  semicircle  ;  if 
.•.  YC  and  ZH  be  produced,  they  will  meet  in  the  circumference  of 
the  circle  at  some  point,  and  YK  will  be  a  diameter.  Hence  YC= 
CK;  and  as  SC=CH,  and  <SCY=<KCH,  the  side  HK  must 
be  equal  to  SY.  But  by  the  property  of  the  circle,  (Euc.  3.  35.) 
ZHxHK=AHxHM=BC2  by  Art.  38.  Hence  (since  HK=SY) 
SYxHZ=BC2. 


That  is,  the  rectangle  of  the  perpendiculars  from  the  foci  upon 
any  tangent,  is  equal  to  the  square  of  the  semi-axis  minor. 


ON   THE    ELLIPSE.  53 

(66.)  Cor.  Since  <s  at  Z  and  Y  are  right  angles,  and  <SPY= 
<HPZ,  the  triangles  SPY,  HPZ,  are  similar  ;  hence, 

SP  :  SY  ::  HP  :  HZ=^-5? ; 

,SYxHZ=SYl^     ; 


or  BC2= 


SP 
SY2xHP 


PS 

SP 

From  which  it  follows,  that  SY2=BC2  x^„  ; 

%  SY=BC  X>/(|^)  «\/(|f)>  BC  being  constant' 


Property  D. 

(66.a.)  Let  the  vertical  tangents  AE,  MF  be  drawn  ;  then 
EA.FM=  BC2  and  EF  is  the  diameter  of  a  circle,  passing  through 
S  and  H. 

If  P  coincide  with  B,  then  E  A  and  FM  each  =BC,  and  E  A.FM= 
BC2.     But  if  not,  let  the  tangent  PE  intersect  the  axis  in  T. 

Then  (sim.  tri.)  EA  :  SY  : :  TA  :  TY, 
and  HZ  :  FM  : :  TZ  ;  TM. 

But  (Euc.  36,  3.  Cor.)  TA.TM=TY.TZ, 
or  TA:TY::TZ:TM; 
.-.  EA  :  SY  : :  HZ  :  FM, 
and  EA.FM=SY.HZ=(65.)BC*. 

Again,  (38.)  AH.HM=BC2=EA.FM, 
.-.  AH  :  EA  : :  FM  :  HM. 

Hence  (Euc.  6.  6.)  the  triangles  EAH  and  HFM  are  similar,  and 
<EHA=<HFM  and  <FHM=AEH.  Whence  <EHF  is  a  right 
angle,  and  a  circle  described  on  EF  will  pass  through  H.  The 
same  may  also  be  shown  of  S. 


54 


ON   THE    ELLIPSE. 


Wherefore,  if  tangents  be  drawn  from  the  vertices  to  meet  any- 
other  tangent,  the  rectangle  of  the  vertical  tangents  will  be  equal 
to  the  square  of  the  semi-minor  axis  ;  and  the  intercepted  part  of 
the  other  tangent  will  be  the  diameter  of  a  circle  passing  through 
the  foci. 

Property  18. 

(67.)  Draw  the  conjugate  diameters  PCG,  DCK,  then  SPx 
HP=CD2. 


By  similar  triangles,  SPY,  HPZ,  PEF,  we  have 


SP 
and     HP 


SY 
HZ 


SPxHP  :  SYxHZ 


PE   :PF, 
PE   :PF 


PE2 :  PF5 


Now,  by  Art.  65,  SYxHZ=BC2, 
Art.  45.  .     .  PE2=AC2 ; 

.-.  SPxHP  :  BC2  : :  AC2 ;  PF2    or  SPxHP- 


BC2xAC2 


PFa 


But  by  Art.  62.    CDxPF=ACxBC ; 


k   ACxBO      ,_■_.    AC2xBC2 
'  CD~- PF~  and  CD  : PF2-" 


Hence  SPxHP=CD2. 


ON   THE    ELLIPSE.  55 

That  is,  the  rectangle  contained  by  the  straight  lines,  drawn  from 
the  foci  to  the  extremity  of  any  diameter,  is  equal  to  the  square  of 
half  the  conjugate  to  that  diameter. 

Property  E. 

(67. a.)  Let  PG,  RX  be  any  two  diameters,  and  let  EF,  parallel 
to  PG,  cut  RX  in  L.     Then  PG2  •  RX2  : :  EL.LF  :  RL.LX. 
Draw  DK  conjugate  to  PG,  and  RN,  Rrc,  ordinates  to  PG,  DK. 


Then  RN=Crc,  HN=CI,  and  Rn=HI=CN. 

By  sim.  tri.  RraC,  CLI,   Crc2  :  Cn2— CP  : :  RC2 :  RC2— • CL2. 

Also  (54.6.)  RN2(Cn2) :  RN2— HN2(Crc2— CP)  : :  PC2— CN* 
:  EI2— HI2(EP— Rrc2). 
But  (sim.  tri.)      C^2  :  Cn2— CI2  : :  Rw2(CN2)  :  Rrc2— LP 

: :  RC2  :  RC2— CL2. 

Our  proportions  then,  are 

Cn2  :  Cn2— CP  : :  PCs— CN*  :  El2— Rn2 

Qrfi  :  Cn2— CP  : :  CN2  :  Rn2— LI2  : :  RC2  :  RC2— CL2 

Adding  the  terms  of  equal ) 
ratios,  by  Euc.  12.  5.      |PC2         :  El2~Ll2  : :  RC2  :  RC2-CL* 

Alternation,  and  Euc.  5.  2,  cor.        PC2  :  RC2  : :  EL.LF  :  RL.LX, 
Or,  PG2  :  RX2  : :  EL.LF  :  QJL.LX. 

The  same  demonstration,  (signs  being  changed  whenever  neces- 
sary,) is  applicable  to  E'F',  which  intersects  RX,  produced,  in  L'. 

Wherefore,  the  squares  of  any  two  diameters  are  to  each  other,  as 
the  rectangles  of  the  segments  of  one  of  them,  are  to  the  rectangles 
of  the  corresponding  segments  of  lines  parallel  to  the  other ;  whether 
the  point  of  intersection  be  within  or  without  the  ellipse. 


56 


ON   THE    ELLIPSE. 


Property  F. 

(67.6.)  Let  PG,  RX,  be  any  diameters,  and  let  EF,  Q,S,  parallel 
to  PG,  RX  respectively,  intersect  each  other  in  M. 

Then  PG2  :  RXB  : :  EM.MF  :  Q,M.MS. 

For,  through  M  draw  the  diameter  AB. 


Then 

AB2 

Or, 

AB2 

In  like  manner, 

AB2 

,« 

.PG2 

PG3  : :  AM.MB  :  EM.MF. 
AM.MB  : :  PG2  :  EM.MF. 
AM.MB  : :  RX2 :  QM.MS. 
RX2::  EM.MF ;  QM.MS. 

Which  demonstration  is  equally  applicable  to  lines  intersecting 
within  or  without  the  ellipse. 

Wherefore,  if  straight  lines  in  the  ellipse  intersect  each  other, 
either  within  or  without  the  curve,  the  rectangles  of  their  corre- 
sponding segments  are  to  each  other  as  the  squares  of  those  diame- 
ters, to  which  they  are  parallel. 

Cor.  When  a  line  becomes  a  tangent,  its  square  corresponds  to 
the  rectangle  in  other  cases.  Therefore,  the  squares  of  tangents 
which  intersect,  are  as  the  squares  of  the  diameters  to  which  they 
are  parallel,  and  the  tangents  themselves  are  as  the  same  diameters. 


These  are  a  few  of  the  most  useful  properties  of  the  Ellipse;  a 
variety  of  others  will  be  found  in  the  Sixth  Chapter,  which  treats 
of  the  analogous  properties  of  the  three  Conic  Sections.  We  now 
proceed  to  the  Hyperbola. 


ON   THE    HYPERBOLA.  57 


CHAPTER  IV. 
<m  THE    HYPERBOLA. 


The  Properties  of  the  Hyperbola  may  be  divided  into  two  class- 
es :  in  the  first  class  may  be  placed  such  properties  as  are  analogous 
to  those  of  the  Ellipse ;  in  the  second  class,  such  as  are  derived 
from  its  relation  to  the  Asymptote.  We  shall  consider  each  of  these 
classes  separately,  beginning  with  that  which  contains  the  properties 
analogous  to  those  of  the  Ellipse. 


VII. 


DEFINITIONS. 

(68.)  Let  PAQ,  be  an  Hyperbola  generated  by  the  revolution  of 
the  lines  SP,  HP  about  the  fixed  points  S,  H,  according  to  the  law 
prescribed  in  Art.  9.  Take  HM=AS,  and  let  the  lines  Sp,  Up,  re- 
volve  round  H,  S,  according  to  the  same  law ;  then  it  is  evident 
that  the  point  p  will  trace  out  another  curve  pMq  passing  through 
M  precisely  similar  to  PAQ,.  pMq  is  therefore  called  the  opposite 
Hyperbola. 

(69.)  The  point  A  is  called  the  vertex  ;  and  the  part  AM  of  the 
line  HS  which  joins  the  two  foci  S  and  H,  is  called  the  Major  axis 
of  the  Hyperbola. 

(70.)  If  AM  be  bisected  in  C,  C  is  called  the  center ;  and  if 
through  C  a  line  BCO  be  drawn  at  right  angles  to  AM,  and  with 
center  A  and  radius  SC  a  circle  be  described  cutting  BCO  in  B  and 
O,  (in  which  case  BC2=AB2— AC2=SC2— AC2),  then  BCO  is  called 
the  Minor  axis  of  the  Hyperbola. 

C.S.  8 


58 


ON    THE    HYPERBOLA. 


(71.)  From  any  point  P  let  fall  the  perpendicular  PN  upon  the 
axis  major  MA  produced,  and  through  the  focus  S  draw  LST  par- 


allel to  it ;  then  PN  is  called  the  ordinate  to  the  axis  ;  AN,  NM,  the 
Abscissas ;  and  the  line  LST  is  called  the  Latus-rectum,  or  the 
Parameter  to  the  axis. 


(72.)  Produce  BCO  both  ways;    take  O,  Gh  equal  to  CS  or 
CH ;  and  with  s,  h  as  foci,  BO  major  axis,  and  AM  conjugate  axis, 


ON   THE    HYPERBOLA.  59 

describe  two  other  hyperbolas  dBD,  KOA: ;  these  are  called  conju- 
gate Hyperbolas.  A  figure  thus  arises  consisting  of  four  Hyperbolas, 
with  their  vertices  A,  B,  M,  O,  turned  towards  each  other,  of  which 
the  opposite  parts  are  similar  and  equal.  If  BCO=ACM,  then 
these  four  Hyperbolas  are  exactly  similar  and  equal ;  and  in  this 
case  the  Hyperbolas  are  said  to  be  Equilateral. 

(73.)  Any  line  PCG  drawn  through  the  center,  and  terminated  by 
the  opposite  hyperbolas,  is  called  a  diameter  ;  the  line  DCK  drawn 
parallel  to  a  tangent  at  P,  and  terminated  by  the  conjugate  hyper- 
bolas, is  called  a  conjugate  diameter  to  PCG.  From  any  point  Q,, 
draw  Q,V  parallel  to  a  tangent  at  P  ;  then  Q,V  is  called  the  ordinate 
to  the  diameter  PCG,  and  PV,  VG  the  abscissas. 

VIII. 

Properties  of  the  Hyperbola  analogous  to  those  of  the  Ellipse. 

Property  1.         (Prop.  2.  of  Ellipse.) 

(74.)  MSxSA=BC2  (See  Fig.  in  p.  58.) 

By  Art.  70.  BC2=SC2  —  AC2. 

=(SC+AC)x(SC— AC). 
=(SC+CM)x(SC— AC)  (for  CM=AC). 
=MSxSA. 

That  is,  the  rectangle  of  the  focal  distances  from  the  vertices,  is 
equal  to  the  square  of  the  semi-axis  minor. 
Cor.  For  the  same  reason,  AHxHM=BC2. 

Property  2.        (Prop.  3.  of  Ellipse.) 

(75.)  The  latus-rectum  LST  is  a  third  proportional  to  the  major 
and  minor  axes.     (Fig.  in  page  58.) 


For 
HL-SL=2AC(by  const.) 

.-.HL=2AC-hSL, 
andHL2=4AC2+4ACx 

SL+SL2. 


Again, 
HL2=SL2+SH2  (Euc.  47.  1.) 

=SL2+4SC2  (SH=2SC), 
=SL2+4AB2, 

=SL2+4(AC2+BC2). 


60  ON    THE    HYPERBOLA. 

Hence  4AC2-HACxSL+SL2-SL2+4AC2-f-4BC2  , 
.-.4ACxSL=4BC2. 
and  2AC  :  2BC  : :  2BC  :  2SL, 
or  AM  :  BO    : :  BO    :  LT. 

Or,  the  latus-rectum  is  a  third  proportional  to  the  major  and  mi- 
nor axes. 

(75.C.)  AS.SM=*LT.AM.    For  (75.)  B02-LT.AM. 
.-.  *B02)=BC2)=(74.)AS.SM=*LT.AM. 

Property  3.        (Prop.  4.  of  Ellipse.) 

(76.)  If  PyT  bisects  the  angle  HPS,  it  will  be  a  tangent  to  the 
Hyperbola  in  P. 

If  PT  be  not  a  tangent,  it  must  cut  the  curve  in  P.  Let  Q,  be 
any  point  within  the  Hyperbola,  in  TP  produced.  Draw  Sys  at 
right  angles  to  TP,  meeting  HP  in  s.  Join  HQ,,  SQ,  sQ.  With 
H  as  center,  and  HQ,  radius,  describe  the  circular  arc  Q,^,  cutting 
the  curve  in  q.  Join  ^S,  qH.  Then,  since  qS,  QS  are  the  bases 
of  the  triangles  ?HS,  QHS  and  <  ?HS>  <  QHS,  ?S>  QS. 
(Euc.  24. 1.) 


In  the  right  angled  triangles  SyP,  syV}  Vy  is  common,  and 
<SPy=<sPy.  .\Sy— sy,  and  Ps=PS.  Also  in  the  right  an 
gled  triangles  QSy,  Qsy,  sy=Sy.  and  Qy  is  common  ;  .-.  Qs=QS. 


ON   THE    HYPERBOLA. 


61 


Since  PS=-Ps,  HP— Ps=HP— PS=AM  ;  i.  e.  Hs=AM.  Also, 
since  QS=Qs,  Ha— as=HO— QS>H?—  qs=AM ;  .-.Hd— Gls> 
E.$.  Or  HQ,>Hs+Q,s  ;  which  (Euc.  20.  1.)  is  impossible.  Hence 
TP  does  not  cut  the  curve,  that  is,  it  touches  it. 

Therefore,  if  from  the  foci  two  straight  lines  be  drawn  to  any 
point  in  the  curve,  the  straight  line  bisecting  the  angle  contained  by 
these  lines,  is  a  tangent. 


(77.)  Cor.  When  P  comes  to  A,  the  <HPS=  two  right  angles  ; 
therefore  a  tangent  at  A  is  perpendicular  to  the  axis  AM. 

Property  4.        (Prop.  5.  of  Ellipse.) 

(78.)  If  tangents  be  drawn  at  the  extremities  of  any  diameter  of 
an  Hyperbola,  they  will  be  parallel  to  each  other. 


62  ON   THE    HYPERBOLA. 

Complete  the  parallelogram  SPHG,  of  which  HP,  PS  are  two 
sides ;  then  since  its  opposite  sides  are  equal,  GS — HG  will  be  equal 
to  HP — PS,  and  by  a  process  similar  to  that  in  the  Ellipse,  (Art.  43.), 
it  may  be  proved  that  G  is  a  point  in  the  opposite  hyperbola,  C  the 
center  of  the  hyperbola,  and  PG  a  diameter. 

Now  in  the  parallelogram  SPHG,  the  opposite  <HGS=<HPS  ; 
but  by  Art.  76,  PT  bisects  the  <HPS,  and  for  the  same  reason  G^- 
bisects  the  <HGS  ;  hence  the  <^GP  is  equal  to  <GPT,  and  there- 
fore ef  is  parallel  to  gh.     Therefore,  if  tangents,  &c. 

(79.)  Hence,  (as  in  the  Ellipse),  if  tangents  be  drawn  at  the  ex- 
tremities of  any  two  diameters  PCG,  DCK,  they  will  form,  by  their 
intersection,  a  parallelogram  eghf. 

Property  5.        (Prop.  6.  of  Ellipse.) 

(80.)  If  SP  and  CD  be  produced  till  they  intersect  each  other  in 
E,  then  PE=AC. 

Draw  HI  parallel  to  CDE  or  e/,  and  produce  SE  to  meet  it  in  I. 
Since  HI  is  parallel  to  P/,  the  exterior  <SP/=interior  <PIH,  and 
<HP/=alternate  <PHI  ;    but    <SP/=<HP/*,  because  PT  bisects 
<HPS  ;    .-.  <PHI=<PIH,  and  consequently  P1=PH.     Again,  be-- 
cause  CE  is  parallel  to  HI,  and  SC=CH,  SE  must  be  equal  to  EI. 

n     HP+PI 
Hence  Pi= — ^ — , 

EI-SI- 

.-.(PI-EI)  or  PE-SltpZ?!, 


HP-PS     2AC 

: - = — A.O. 


Hence,  if  through  the  extremity  of  any  diameter,  a  line  be  drawn 
from  the  focus,  to  meet  the  conjugate  diameter  produced,  the  part 
intercepted  by  the  conjugate  will  be  equal  to  the  semi-axis  major. 


ON    THE    HYPERBOLA.  63 


Property  6.         (Prop.  7.  of  Ellipse.) 

(81.)  If  the  ordinate  PN  be  drawn  to  the  major  axis,  then  ANx 
NM  :  PN2  : :  AC2 :  BC2.     (Fig.  in  p.  64.) 


Let 
AC  or  CM=a, 


fijr^ 


Then,  by  Art.  9. 


SC  or  CR=b,  )       (b*— a?)x(x— a)x(x+a)=a?y\ 


CN=;r, 
PN=y; 


or  (SC2— AC2)  x  AN  x  NM=AC2  x  PN2. 
But,  by  construction,  SC2— AC2=BC2 ; 

.-.  BC2xANxNM=AC2xPN2, 
and  ANxNM  :  PN2  : :  AC2  :  BC2.* 


Therefore,  as  the  square  of  the  major  axis  is  to  the  square  of  the 
minor,  so  are  the  rectangles  of  the  abscissas  of  the  former,  to  the 
squares  of  their  ordinates. 


Cor.  1.  Si^  ANxNM=(CN— AC)x(CN+AC)=CN2— AC5 
CN2-— AC2  :  H2  : :  AC2  :  BC2. 


(82.)  Cor.  2.  Produce  NP  to  p,  and  draw  any  ordinate  pm  at 
t  angles  to  Cm,  then  (since  the  conjugate  hyperbola  Bp  is  de- 
d  with  BC  as  major  and  AC  minor  axis)  BmxmO  :  pm2  : : 
AC2,  or  (since  BmxmO=(Cm— BC)x(Cm+BC))  Cm2— BC2 
:  pm2  : :  BC2  :  AC2. 


[pa. 

fright  i 


(83.)  Cor.  3.  Since  7?N=Cm,  and  pm=CN,  we  have  (by  Cor.  2.) 
pN2— BC2  :  CN2  : :      BC2        :  AC2, 
or  pN^-BC2  :  BC2  : :       CN2        :  AC2, 
anddividendo,pN2-2BC2  :  BC2  ::  CN2— AC2  :  AC2, 

: :      PN*     :  BC2. 


*  The  general  property  of  the  Hyperbola  analogous  to  the  10th 
Property  of  the  Ellipse,  viz.  VvxvG  :  Gh?2  : :  PC2  :  CD2,  will  be 
found  at  the  end  of  the  Properties  of  the  Hyperbola  derived  from  its 
relation  to  the  Asymptote. 


64 


ON   THE    HYPERBOLA. 


Hence  pN2— 2BC2=PN2,  and  />N2— PN2  =  2BC2 ;  in  the  same 
manner,  if  nip  be  produced  to  q,  it  may  be  proved  that  qm2- 
pm*=2AC\ 

That  is,  the  square  of  any  ordinate  to  either  axis  is  less  than  the 
square  of  the  same  ordinate  produced  to  the  conjugate  Hyperbola,  by 
twice  the  square  of  the  semi-axis,  to  which  it  is  parallel. 

Property  7.        (Prop.  8.  of  Ellipse.) 

(84.)  If  the  ordinate  Vn  be  drawn  to  the  minor  axis,  then  BC2-}- 
Cn2  :  Prc2  : :  BC2  :  AC2. 


In  this  case,  Prc=CN,   and   C/i  =  PN;   therefore  ANxNM 


:CN, 

.CTa 


(CN2— AC2=)Prc2— ACT  and  PN2  =  Cti2 ;  hence,  by  substitution  in 
Art.  81,  Cor.  1,  we  have, 


P^—AC2  :  Cw2  : 

.-.  Prc2— AC2  :  AC2 : 

Compone?ido,  Pra2  :  AC2 : 

or  BC2+Cw2  :  Pn2  : 


AC2  :BC2; 

Crc2  :  BC2. 

BC2+Cw2  :BC2, 

BC2  :  AC2. 


ON   THE    HYPERBOLA.  65 

That  is,  as  the  square  of  the  minor  axis  is  to  the  square  of  the 
major,  so  is  the  sum  of  the  squares  of  the  semi-minor,  and  of  the 
distance  from  the  center  to  any  ordinate  upon  the  minor,  to  the 
square  of  that  ordinate. 

Property  8.        (Prop.  11.  of  Ellipse;)       % 

(85.)  If  the  tangent  PT  cuts  the  major  axis  in  T,  and  the  minor 
axis  in  t,  then  CNxCT=AC2,  and  C*xCrc=BC2.     (See  last  Fig.) 

Since  PT  bisects  the  angle  UPS,  by  Euc.  3,  6.  we  have 

HT:  TS  ::  HP  :PS; 

.-.  HT— TS(2CT)*  :  HT+TS(SH) : :  HP— PS(2AC) :  HP+PS.   (A) 
But  by  Euc.  12.  2,  HP2=HS2+  PS2+2HSxSN  ; 
.-.  HP2— PS2=HS2+2HSxSN, 

=(HS+SN)2— SN2, 
=HN2— SN2 ; 
.-.  HN— SN(a^HP— PS(2AC)  : :  HP+PS  ;  HN+SN(2CN).|  (B) 

Hence  we  have,  /    §  / 

I  : :     2AC     :  HP+PS,  (A)   tH^kY>^mj 


• 


SH  :  2AC 
.  2CT  :  2AC 
d#CT  :   AC 


HP+PS  :  2CN  ;    (B) 

2AC     :  2CN, 

AC      ;  CN,  or  CNxCT=AC2. 


C*t    \  (  C**CTk  A 


AC        :  CT    (and  first  :  third 

CN2      :  CA2 ; 
CN2— AC2  :  AC2  ::  PN2  :  BC2.    <- N\  tj- 
But  bv  sim.  As,     PTfr,    TC*,    NT  :  CT    : :  PN    :  Ct. 
Hence,     PN  :  Ct  : :  PN2  :  BC2,    or    PNxC*=BC2, 
but  PN=Crc,  .-.  C?*xC*=BC2. 


(86.)   Since    CN  :  AC 

first2  :  second2)  CN  :  CT 
dividendo,  NT  :  CT 


*  For  HT— TS=HC+CT— T%=SC+CT— (SC-CT)=2CT. 
t  For  HN+SN=HS+2SN=2CS+2SN=2(CS+SN)=2CN. 


c.s. 


66 


ON   THE    IIYPERBOLA. 


Therefore,  if  a  tangent  and  ordinate  be  drawn  from  any  point  of 
the  curve  to  either  of  the  axes,  half  that  axis  will  be  a  mean  pro- 
portional between  the  distances  of  the  two  intersections  from  the 
center. 

(87.)   Coi^CN2— AC2=CN2— CN  xCT=CNx(CN— CT)=CN  x 

NT. 

Property  A.        (Prop.  C.  of  Ellipse.) 

(87. a.)  Let  TLG  be  the  focal  tangent,  or  the  tangent  drawn  at 
the  extremity  of  SL,  the  ordinate  from  the  focus.  Let  NPG  be 
any  ordinate,  produced  to  meet  the  tangent  TLG.     Then  SP=NG. 


*» 


If  AI  be  taken  equal  to  SP,  then  IM=HP.     z   '''  1    *SZr  n  *h 
".•  SP=Cf-CA,  and  HP=CI+CA.  X  PX*  P  H  V  *  J^ 

-   (Euc.  47.  h)  (SP2)(CI— CA)2=PN2+(CN— CS)2(NS2).    H?  * 
^  And  (HP2)(CI+CA)2=PN2+(C^+CS)2(NH2). 

fexjmnd  and  subtract        4CI.CA=4CN.CS,  and        ^  ?  y  C\« 
\  %    CI.CA=CN.CS.      .-.  CN  :  CI(SP+AC)  : :  CA  :  CS 

: :  CT  :  CA  (85.) 

...  CN— CT(TN)  :  SP  : :  CT  :  CA.     (Euc.  19.  5.) 

Again,  CS.CT=AC2,  and  (70.)  CS2— AC2(=CS.ST)=BC2. 
.-.  CS.CT  :  CS.ST,  or  CT  :  ST  : :  AC3  :  BC2 

: :  AC   ;  SL  (75.) 
.-.  ST  :  SL  : :  CT  :  AC  : :  TN  ;  SP. 


ptJ 


i  s 


P+*£~ 


A  C 


r  *4 


/■ 


tC^c 


m* 


/AS 


' 


ON   THE    HYPERBOLA.  67 

But  (sim.  tri.)        ST  :  SL  : :  TN  :  NG. 
.-.  SP=NG ; 

That  is,  the  distance  from  the  focus  to  any  point  of  the  curve  is 
equal  to  the  ordinate  to  that  point,  produced  until  it  meets  the  focal 
tangent. 

Cor.  1.  AS=AE  and  SM=MF.     Also  C£=AC. 
For  (sim.  tri.)  CT  :  C*  : :  ST  :  SL  : :  CT  :  AC. 

Cor.  2.  Hence*  also,  since  TN  :  NG  is  a  constant  ratio,  TN  : 
SP  is  a  constant  ratio.  •Therefore,  if  a  line  be  drawn  through  T  per- 
pendicular to  AC,  the  distance  of  the  point  P  from  that  line  (=TN) 
is  in  a  constant  ratio  to  SP,  the  distance  of  the  same  point 'from 
the  focus.  This  ratio,  being  (by  demonstration  above)  =CT  :  CA, 
is  a  ratio  of  less  inequality.  This  perpendicular  is  the  directrix  of 
the  Hyperbola.     (See  Art.  138,  et  seq.) 

Property  9.         (Prop.  12.  of  Ellipse.) 

(88.)  If  PCG,  DCK,  be  conjugate  diameters  of  the  Hyperbola, 
and  OPF  be  drawn  perpendicular  to  CD  produced  *if  necessary, 
then  POxPF=BC2.     (See  next  Fig.) 


all^|  < 
POX,  i 


Draw  Cy  parallel  to  PF.  Then  because  PO  is  parallel^  Cy, 
and  Ct  parallel  to  PN,  the  right-angled  triangles  tCy,  PON7  are 
similar  ;  .-.  Ct  :  Cy  : :  PO  :  PN.  But  Cy=PF,  and  PN=Crc,  being 
opposite  sides  oQk  parallelogram  ; 

.-.  C* :  PF  : :  PO  ;Crc,  or  POxPF=C*xC?z=BC2  (86.) 

Therefore,  if  from  the  extremity  of  any  diameter,  a  perpendicular 
i  ,  w  drawn  to  its  conjugate  ;  then  the  rectangle  of  that  perpendicular 
and  the  part  of  it  intercepted  by  the  axis  major,  will  be  equal  to  the 
square  of  the  semi-axis  minor. 

Property  10.         (Prop.  13.  of  Ellipse.) 

(89.)   Draw  the  ordinates   DL,   PN,   to  the   major   axis,  then 
CN2— CL2=AC2,  and  DL2— PN2=BC2.     (See  next  Fig.) 


68 


ON    THE    HYPERBOLA. 


•  Draw  the  ordinate  mD,  and  produce  it  to  meet  the  hyperbola  in 
q,  and  draw  qk  perpendicular  to  the  major  axis. 


By  Cor.  1.  Art.  81.    Or2-— AC2 :  kq% : 

and    CN2— AC2 :  PN2  : 

.-.C&2— AC2:   kf\ 


AC2        :  BC2, 

AC2        :  BC2, 

CN2— CA2  :  PN2. 


\/c=mqj  mD=CL,  and  qk=T>Ij ;  and  by  Art.  83.  mq2- 
m*  JI^AC2;  .-.^2— AC2=(CA;2~AC2=)AC2-fmD2=AC2+CLs 
hence,  by  substitution,  we  have, 

AC2+CL2  :  DL2  : :  CN2— AC2 :  PN2     (A.) 


By  sim.  As,  DCL,  PTN, 

/   1/4***]  DL2:CL2 

aequo,  AC2-f  CL2  ♦  CL2 


Hence  AC2-f  CL2 :  AC2 


PN2        :  NT2, 
CN2— AC2  :NT2, 
CNxNT    :NT2,     (87.)  ffl 
CN        :  NT. 

:  CN— NT(CT) 
:CTxCN, 
:  AC2    (85.) 


ON    THE    HYPERBOLA. 


69 


.-.  AC2+CL2=CN2,     j. 
or  CN2— CL2=AC. 

(90.)  Since  AC2+CL2=CN2,  by  substitution  in  proportion  (A), 
we  have, 


CN2 :       DL2       : 

or  CN2 :  CN2— AC2  : 

and  divd0,  AC2  :  CN2— AC2  : 

.-.  AC2  :  DL2— PN2  : 


CN2— AC2 :  PN2, 

DL2       :  PN2, 

DL2— PN2 :  PN2, 

CN2— AC2 :  PN2, 

AC2       :  BC2 ; 


-,&* 


.-.  DL2— PN2=BC2. 

Hence,  if  ordinates  to  either  axis  be  drawn  from  the  extremities 
of  any  two  conjugate  diameters,  the  difference  of  their  squares  will 
be  equal  to  the  square  of  half  the  other  axis. 


Property  11.        (Prop.  14.  of  Ellipse.) 

(91.)  PC2^CD2=ACVBC2, 
PG  and  DK  being  conjugate  diameters. 

For  by  Arts.  89,  90.  CN2— CL2=AC2, 
and  DL2— PN2=BC2, 

.-.  CN*-f  PN2— (CL2+DL2)=AC2— BC2, 
or  CP2— CD2=AC2— BC2. 


Hence  the  din%§ence  of  the  squares  of  any  two  semi-conjugate  di- 
ameters is  equal  to  the  difference  of  the  squares  of  the  semi-axes. 

Property  12.        (Prop.  15.  of  Ellipse.) 
CDxPF-ACxBC. 


Tn  Art.  90.  it  was  proved  that 
CN2  :  DL2  : 
.-.CN   :DL   : 
or  CN  :AC   : 


AC2  :  BC2 ; 
AC  :BC, 
DL  :  BC. 


70  ON   THE    HYPERBOLA. 

But  by  sim.  *s,  TCy,  DCL,   CT  :  Cy(PF) : :  CD  :  DL. 


P 


Hence  we  have,    CN  :        AC 
and    CT:         PF 

.-.  CNxCT(AC2)  :   ACxPF 
or    AC:    '     PF 


fr-*1****     i  .-.  CDxPF=ACxBC. 


DL  :  BC, 
CD  :  DL ; 
CD  :  BC, 
CD:BC; 


That  is,  if  from  the  extremity  of  any  diameter,  a  perpendicular 
be  drawn  to  its  conjugate,  the  rectangle  of  that  perpendicular  and 
the  semi-conjugate,  is  equal  to  the  rectangle  of  the  semi-axes. 

(92.)  Cor.  Hence  it  appears,  that  all  the  parallelograms  inscri- 
bed in  the  Hyperbolas,  and  having  their  sides  drawn  through  the  ex- 
tremities of  any  diameter  and  its  conjugate,  are  equal  to  each  other 
and  to  the  parallelogram  described  about  the  major  and  minor  axes  ; 
for  the  parallelogram  eghf  (see  Fig.  in  page  61.)  described  about 
the  conjugate  diameters  PCG,  DCK,  is  equal  to  four  times  eDCP= 
4CD  X  PF=4AC  x  BC=  right-angled  parallelogram  whose  sides  are 
2AC  and  2BC=  parallelogram  described  about  the  major  and  minor 
axes. 


Property  13.        (Prop.  16.  of  Ellipse.) 

(93.)  If  SY,  HZ,  be  perpendiculars  drawn  from  the  foci  to  the 
tangent  PYZ,  then  the  points  Y  and  Z  are  in  the  circumference  of 
circle  described  upon  the  major  axis  AM. 

Join  YC,  and  produce  SY  to  meet  HP  in  W. 

Since  the  tangent  PYZ  bisects  the  <  HPS  ;  in  the  right-angled 
triangles  WPY,  SPY,  we  shall  have  PW=PS,  and  WY=YS. 
Now  since  WY=YS,  and  HC=CS,  CY  must  be  parallel  to  HW, 
and  .-.  SC  :  SH  : :  CY  :  HW  ;  but  SC=|SH  ;  .-.  CY=£HW=4(HP 
— PW)=^(HP— PS)=4AM=AC  ;  .-.  Y  is  a  point  in  the  circle  whose 
center  is  C,  and  radius  C A.  In  the  same  manner  it  might  be  proved 
that  Z  is  a  point  in  the  same  circle. 


ON   THE    HYPERBOLA. 


71 


Hence,  if  perpendiculars  be  dropped  from  the  foci  upon  any  tan- 
gent to  the  hyperbola,  the  intersections  of  those  perpendiculars  with 


the  tangent  will  be  in  the  circumference  of  a  circle  described  upon 
the  axis  major. 

Property  14.         (Prop.  17.  of  Ellipse.) 

(94.)  SYxHZ=BC2. 

Since  the  <HZP  is  a  right  angle,  it  must  be  in  a  semicircle  ;  if 
.-.  YC  is  produced  to  meet  HZ  in  K,  K  will  be  in  the  circumference 
of  the  circle,  and  YK  will  be  a  diameter.  Hence  YC=CK ;  and 
as  SC=CH,  and  <SCY=<KCH,  the  side  HK  must  be  equal  to  SY. 
By  the  property  of  the  circle  (Euc.  36.  3.)  HKxHZ=HMxHA=BC2, 
(74.)    Hence  (since  HK=SY)  SYxHZ=BC2. 

That  is,  the  rectangle  of  the  perpendiculars  from  the  foci  upon 
any  tangent  is  equal  to  the  square  of  the  semi-axis  minor. 


/ 


72  ON   THE    HYPERBOLA. 

(95.)  Cor.  By  sim.  As,  SPY,  HPZ, 

SYyHP 
SP  :  SY  ::  HP  :  HZ=       *        ; 

a  Y2  v  ttp 

.-.SYxHZ-      gp— -B0». 

op 

'      and    SY2=BC2Xgp; 

SP\         //SP 


•••^-V(w)V(S> 


Property  B.        (Prop.  D.  of  Ellipse.) 

(95.a.)  Let  the  vertical  tangents  AE',  MF'  be  drawn  ;  then 
E'A.F'M=BC2,  and  E'F'  is  the  diameter  of  a  circle  passing  through 
S  and  H. 

By  sim.  tri.  E'A  :  SY  : :  TA  :  TY, 
and    HZ:FM::TZ:TM. 
But  (Euc.  35.  3.)    TA.TM=TY.TZ. 
Or,  TA  :  TY  : :  TZ  :  TM. 
.-.E'A:  SY::HZ  :  F'M, 
and    E'A.F'M=SY.HZ=(94.)BC2. 

:      Again,  (74.)    AH.HM=BC2=E'A.F'M, 
.-.  AH  :  E'A  : :  PI :  HM. 

Hence  (Euc.  6.  6.)  the  triangles  E'AH  and  HF'M  are  similar,  and 
<E'HA=<HF'M  and  <F'HM=<AE'H.  Whence  <E'HF*  is  a 
right  angle,  and  a  circle  described  on  E'F'  will  pass  through  H. 
The  same  may  also  be  shown  of  S. 

Wherefore,  if  tangents  be  drawn  from  the  vertices  to  meet  any 

other  tangent,  the  rectangle  of  the  vertical  tangents  will  be  equal 

to  the  square  of  the  semi-axis  minor  ;  and  the  intercepted  part  of 

the  other  tangent  will  be  the  diameter  of  a  circle  passing  through 

f      the  foci. 

*  The  points  EH,  HF'  should  be  joined  in  order  to  form  the  tri- 
angles E  AH,  HMF'. 


ON   THE    HYPERBOLA.  73 


Property  15.         (Prop.  18.  of  Ellipse.) 

(96.)  Draw  the  semi-conjugate  diameter  CD,  then   SPxHP= 
CD2. 


By  sim.  As,    SPY,    HPZ,    PEF, 


SP 
and     HP 


.-.  SPxHP 


SY 
HZ 


PE        ;  PF, 
PE        :PF 


SYxHZ(BC2) 

.-.  SPxHP 


PE2(AC2)  :  PF2 ; 
AC2xBC2 
PF2     * 
But  by  Property  12.     CDxPF=ACxBC  ; 

Hence  SPxHP=CD2. 

Or,  the  rectangle  contained  by  the  straight  lines  drawn  from  the 
foci  to  the  extremity  of  any  diameter,  is  equal  to  the  square  of  half 
the  conjugate  to  that  diameter. 

IX. 

On  the  Properties  of  the  Hyperbola  derived  from  its  relation  to 

the  Asymptote. 

The  properties  of  the  Hyperbola  hitherto  exhibited  are  perfectly 
analogous  to  those  of  the  Ellipse  ;  we  proceed  now  to  explain  some 
of  the  properties  in  which  these  two  curves  essentially  differ.  But 
we  must  first  show  what  is  meant  by  an  Asymptote. 

(97.)  Since  the  two  branches  of  the  opposite  hyperbolas  are  pre- 
cisely equal  and  similar  on  each  side  of  the  axis  LK,  if  two  ordinates 
PCI  pa  be  drawn  at  equal  distances  AN,  Mm,  from  the  points  A,  M, 
then  the  tangents  to  the  points  P,  Q,  will  meet  in  the  same  point  T, 
and  tangents  to  the  points  p,  q  in  the  same  point  t.  Now  by  Art.  85, 
CNxCT=AC2,  and  since  AC  is  a  constant  quantity,  CT  varies  in- 
versely as  CN  ;  when  CN  therefore  becomes  infinite  CT  will  be 

C.  S.  10 


74 


ON    THE    HYPERBOLA. 


equal  to  0  ;    i,  e.  if  P,  Q,,  are  points  in  the  curve  at  an  infinite  dis- 
tance, the  tangents  PT,  Q/F  will  meet  in  C  ;*  for  the  same  reason 


-K 


if  p,  q  are  points,  at  an  infinite  distance  in  the  opposite  hyperbola, 
then  the  tangents  pt,  qt  will  also  meet  in  O  ;  and  since  the  <PTQ,= 
<iptq,  these  four  lines  will  evidently  then  coalesce  into  two,  viz.  PT 
with  tq  and  pt  with  TQ,.  The  tangents  to  the  two  opposite  hy- 
perbolas at  an  infinite  distance,  may  therefore  be  represented  by  two 
lines  XCZ,  UOY,  intersecting  each  other  in  C,  and  making  equal 
angles  XCK,  UCL,  KCY,  ZCL  with  the  axis.  These  lines  XCZ, 
UCY  are  called  Asymptotes  ;  and  we  are  now  to  determine  their 
position  with  respect  to  the  axes  of  the  hyperbolas. 


*  To  make  this  more  intelligible,  conceive  PN  to  move  parallel 
to  itself  in  the  direction  NK,  then  since  CNxCT=  a  constant  quan- 
tity, whilst  CN  varies  through  ail  degrees  of  magnitude,  the  point 
T  will  only  pass  from  T  to  C  so  as  to  make  CT=0  ;  i.  e.  when  P 
is  a  point  in  the  curve  at  an  infinite  distance,  the  tangent  PT  will 
pass  through  C  :  and  so  of  the  rest. 


ON    THE    HYPERBOLA.  75 

(98.)  Draw  Aa  at  right  angles  to  AM.  When  P  is  removed  to 
an  infinite  distance,  the  triangle  PNT  becomes  similar  to  the  triangle 
a  AC,  and  CN  becomes  the  same  as  NT.  Hence,  in  this  case, 
PN  :  NT  or  NC  : :  Aa  :  AC  (A);  but  by  Cor.  1,  Art.  81,  CN2— 
AC2  :  PN2  : :  AC2  :  BC2 ;  and  when  CN  is  infinite,  AC  vanishes 
with  respect  to  CN,*  therefore  this  latter  proportion  becomes 
CN2  :  PN2  : :  AC2  :  BC2,  or  CN  :  PN  :  :  AC  :  BC  (B) ;  com- 
pare the  two  proportions  (A)  and  (B),  and  we  have  Aa  :  AC  : : 
BC  :  AC,  or  Aa=BC.  Draw  therefore  Aa  at  right  angles  to  AM, 
and  make  it  equal  to  BC,  join  Ca,  and  this  gives  the  position  of  the 
asymptote  XCZ.  In  the  same  manner,  by  making  A6=BC,  and 
joining  C6,  we  determine  the  position  of  the  asymptote  UCY  ;  in- 
deed, from  what  has  been  proved,  it  appears,  that  if  a  parallelogram 
acdb  be  described  about  the  major  and  minor  axes,  the  asymptotes 
will  be  merely  the  prolongation  of  the  diagonals  of  such  parallelo- 
gram. 

(99.)  These  lines  XCZ,  UCY,  will  also  be  asymptotes  to  the  con- 
jugate hyperbolas  ;  for  by  a  similar  process  of  reasoning  it  might  be 
shown  that  the  position  of  their  asymptotes  would  be  determined 
by  drawing  perpendiculars  Ba,  06,  at  B  and  O,  and  making  Ba  and 
06  each  equal  to  AC.  Thus  these  four  hyperbolas  are  inclosed  as 
it  were  between  their  asymptotes  ;  and  by  producing  the  ordinates 
to  meet  these  asymptotes,  new  properties  of  the  curves  will  arise, 
which  we  shall  now  proceed  to  investigate. 


*  To  show  that  in  this  case  CN2 — AC2  may  be  considered  as 
equal  to  CN2,  let  CA=a,  AN=:r,  then  CN=^+a,  and  CN2= 
z*  _|_  2ax  +  a2  ;  hence  CN2— AC2(  =  CN2— a2)  =  x2  +  2ax  ;  we 
have  therefore  CN2  :  CN2— AC2  : :  x%  +  2a*+a2  :  &  +Hq*  n  *+ 

2a +—  :  x-\-2a\  but  when  x  is  infinite,       becomes  equal  to  0;  in 

x  x 

this  case,  therefore,  this  latter  ratio  becomes  a  ratio  of  equality,  from 
which  it  follows  that  CN2  may  be  substituted  for  CN2— AC2. 


76 


on  the  iiyperbloa. 
Property  16. 


(100.)  Let  the  ordinate  Vp  be  produced  to  meet  the  asymptotes 
in  the  points  L,  I ;  then  PL.PZ=BC2  and  />Z.pL=BC2 ;  also  PL.L/)= 
BC2  and  pZ.ZP=BC2. 

By  Cor.  1.  Art.  81, 

CN2— CA2  :  PN2  : :  AC2 :  BC2. 


By  sim.  As,  LNC,  aAC, 

CN2  :  LN2  : :  AC2  :  Aa2  (BC2) 


...  ON2  :  CN2— CA2  : 

and  dividendo,    AC2  :  CN2— CA2  : 

or    AC2  :  LN2— PN2  : 


LN2  :  PN2, 
LN2— PN2 :  PN2 ; 
ON2— CA2 :  PN2, 

AC2 :  BC2  i 


...  LN2— PN2=BC2. 

But  LN2— PN2=(LN— PN)  x  (LN+PN)=PLxPZ 
.-.  PLxPZ=BC2  or  PL.Lp=BC2. 
For  the  same  reason,  p/xpL=BC2  or  ^/./P=BC2. 


ON    THE    HYPERBOLA.  77 

Therefore,  if  an  ordinate  to  the  axis-major  be  produced  to  meet 
the  asymptotes,  then  the  rectangle  of  the  segments  intercepted  be- 
tween the  curve  and  either  asymptote  will  be  equal  to  the  square  of 
the  semi-axis  minor. 

(101.)  Cor.  1.  Hence  VLxP  l=plxpL=YL.Lp=pl.lP. 

Cor.  2.  Draw  any  other  ordinate  Qq,  and  produce  it  to  meet  the 
asymptotes  in  X  and  Y,  then  will  Q,XxQY=Aa2 ;  hence  we  have 
QXxGlY=PLxPZ. 

Property  17. 

(102.)  Draw  any  diameter  PCG,  and  produce  it  tog)  draw  the 
ordinate  Q,T  to  that  diameter,  and  produce  it  to  meet  the  asymptotes 
R,  r ;  then  QRxQr=TrxTR.     (See  last  Fig.) 

Through  the  points  P,  Q,,  draw  hi,  XY  perpendicular  to  the  axis 
of  the  hyperbola,  and  draw  the  tangent  ef  at  P. 

By  sim.  triangles  Q,XR,  PLe  ;  QrY,  Vfl  ;  we  have 
aX  :       aR     ::     PL      :      Pe, 

andqY  :__<^ : :     PI       t Vf) 

.-.  Q,XxQ,Y  :  QRxQr  : :  PLxPZ  :  PexP/. 

But  by  Art.  101,  aXxdY=PLxPZ ;  hence  QRxQr=PexP/. 

In  the  same  manner,  by  drawing  an  ordinate  through  T  perpen- 
dicular to  the  axis,  it  might  be  shown  that  TrxTR=PexP/;  hence 
QRxQr=TrxTR. 

Therefore,  if  an  ordinate  to  any  diameter  be  produced  to  meet 
the  asymptotes,  the  rectangle  of  the  segments  intercepted  between 
the  curve  and  one  asymptote,  will  be  equal  to  the  rectangle  of  the 
segments  intercepted  between  the  curve  and  the  other. 

(103.)  Cor.  1.  Since  Q,r=Q/T  +  Tr,  and  TR=Q,T-f-QR,  we 
have 

aRx(dT-fTr)       =Trx(QT+QR), 
or  QRxaT+QRxTr=TrxQ,T+TrxQR. 

Subtract  QRxTr  from  each  side  of  this  latter  equation,  and  there 
results  QRxQ,T=TrxQ,T,  from  which  it  appears  that  QR=Tr ; 


78  ON   THE    HYPERBOLA. 

in  the  same  manner  it  may  be  proved  that  Q,X=</Y,  and  PL=//Z. 
If  therefore  Rr  moves  parallel  to  itself  till  it  comes  into  the  position 
of  the  tangent  ef  (in  which  case  the  points  Q,  and  T  coincide  in  P), 
we  shall  have  Pe=P/",  and  consequently  PexP/=Pe2. 

(104.)  Cor.  2.  Since  the  triangles  eCf  RCr  are  similar,  and 
since  the  diameter  GCg  bisects  ef  in  P,  it  will  bisect  Rr  in  V ; 
hence  VR^=Vr  ;  and  as  Q,R=Tr,  we  have  YQ,=VT,  i.  e.  the  diame- 
ter GOg-  bisects  all  its  ordinates. 

(105.)  Cor.  3.  Hence  YR2— VQ2=Pe2.  For  VR2— VQ2=(YR— 
YQ)  X  (VR-t-VQ)=(VR— YQ)  x  (VR+YT)  =  QRxRT  =  QR  xQr, 
(for  RT=Q,R-hQT=Tr+QT==Qr).  But  QRxQr=PexP/=(103) 
Pe2;  .-.  VR2— YQ2=Pe2. 

Property  18. 

(106.)  Join  AB,  and  let  it  cut  the  asymptote  XCZ  in  S ;  draw 
PD  parallel  to  the  asymptote  UCY,  cutting  the  asymptote  XCZ  in 
R  ;  then  CRxRP=AS2.     (Fig.  in  next  page.) 

Since  the  diagonals  BA,  aC  of  the  parallelogram  aBCA  are  equal 
and  bisect  each  other  in  the  point  S,  the  lines  SA,  SC,  SB,  Sa  are 
equal  j  hence  the  <  SAC  =  <  SCA  ;  but  <  SCA  =  <  ACY,  .-. 
<SAC=<ACY,  and  consequently  AB  is  parallel  to  UCY.  If 
therefore  Pr,  Am  are  drawn  parallel  to  the  asymptote  XCZ,  then 
PRCr,  ASCm  will  be  parallelograms,  and  Pr  will  be  equal  to  CR, 
and  Am  to  SC. 

By  sim.  triangles  PrJ,  Amb  ;  PRL,  AS«  ;  we  have 


Pr(CR)  :      VI 

and  RP  :      PL 


.-.  CRxRP  :  PLxPZ 


Am(SC)  :    A6, 
SA     :     Aa : 


SC  x  SA  :  AaxAb. 


But  PLxPZ=AaxA6  ;  hence  CRxRP=SCxSA=SA2. 

That  is,  if  from  any  point  of  the  curve  a  line  be  drawn  to  the 
nearer  asymptote,  parallel  to  the  other  asymptote,  the  rectangle  of 
this  line,  and  the  distance  of  its  intersection  with  the  asymptote  from 
the  center,  is  a  constant  quantity ;  and  is  equal  to  the  square  of  half 
the  diagonal  of  the  rectangle  of  the  semi-axes. 


ON    THE    HYPERBOLA. 


79 


(107.)  Cor.  1.  Since  XCZ  is  likewise  an  asymptote  to  the  con- 
jugate hyperbola,  by  a  similar  process  of  reasoning  it  might  be  shown 
that  CRxRD=SB2=SA2 ;  hence  CRxRD=CRxRP,  and  conse- 
quently RD=RP,  i.  e.  PD  is  bisected  by  the  asymptote. 

(108.)  Cor.  2.  Since  SA=4AB,  SA2  is  a  constant  quantity ; 
hence  RP  varies  inversely  as  CR ;  when  CR  therefore  is  infinite, 
HP  will  become  equal  to  0  ;  which  coincides  with  what  has  already 
been  said  as  to  the  asymptote's  touching  the  curve  at  an  infinite 
distance. 


Property  19. 

Join  CD,  and  produce  it  to  K ;  draw  the  diameter  PCG  ;  then 
will  DCK  be  the  conjugate  diameter  to  PCG.     (Fig.  in  next  page.) 


(109.)  Draw  ef  touching  the  curve  in  the  point  P,  and  meeting 
the  asymptotes  XCZ,  UCY  in  the  points  e  and/;  then  by  Art.  103, 
P/  will  be  equal  to  Pe  ;  and  since  PR  is  parallel  to  C/;  CR  will  be 


80 


ON    THE    HYPERBOLA. 


also  equal  to  Re.  Hence,  in  the  triangles  CRD,  PRe,  we  have 
CR=Re,  RD=RP,  and  <CRD=<eRP,  .-.  (Euc.  4.  1.)  CD  is  equal 
to  Pe,  and  the  <DCR  equal  to  the  <ReP;  consequently  DCK  is 
parallel  to  the  tangent  ef,  and  is  therefore  the  conjugate  diameter  to 
PCG  (73.) 


Therefore,  if  a  parallel  to  either  asymptote  cut  conjugate  Hyper- 
bolas, the  diameters  passing  through  the  points  of  intersection  will 
be  conjugate  to  each  other. 

(110.)  Cor.  Join  Be,  then  eDCP  will  be  a  parallelogram,*  whose 
diagonal  is  Ce  ;  and  as  Be  is  parallel  to  the  diameter  PCG,  it  touches 
the  conjugate  hyperbola  in  D.     Complete  the  parallelogram  eghf, 


*  For  CD  being  equal  and  parallel  to  Pe,  De  must  be  equal  and 
parallel  to  CP.     (Euc.  33.  1.) 


ON   THE    HYPERBOLA.  81 

as  in  Art.  79,  then  in  the  same  manner  as  it  has  been  proved  that 
Ce  is  the  diagonal  of  the  parallelogram  eDCP,  it  might  also  be  prov- 
ed that  the  point  h  would  be  found  in  the  asymptote  XCZ,  and  the 
points  g,  f  in  the  asymptote  UCY ;  these  asymptotes  are  therefore 
the  prolongation  not  on!  y  of  the  diagonals  of  the  parallelogram  de- 
scribed about  the  major  and  minor  axes,  but  also  of  the  parallelo- 
gram described  about  any  two  conjugate  diameters. 

Property  20.         (Prop.  10.  of  Ellipse.) 

(111.)  Draw  the  ordinate  Q,v,  then  FvxvG  :  Q,r2  : :  PC2  :  CD2  : : 
PG2  :  DK2. 

5 

Produce  vQ,  to  X,  then,  by  sim.  As,  OX,  CPe, 
Cv2  :       CP2      ::       vX2       :     Pe2; 

...  (V—  CP2  :       CP2      : :  vX2— Pe2  :     Pe2, 
and  Cv2— CP2  :  vX2— Pe2 : :       CP2       :     Pe2.  (A) 

But  Ct?»— CP2=(0— CP)  x  (Cv+CV)=PvxvG. 
By  Art.  105.  vX2— vQ,2=Pe2,    .-.  vX2   — Pe2    =vQ,2. 

Now  (109.)  Pe2=CD2. 
Hence,  by  substitution  in  Proportion  (A),  we  have 
FvxvG  :  Qv2  : :  CP2  :  CD2  : :  PG2  :  DK2. 

Therefore,  the  square  of  any  diameter  is  to  the  square  of  ils  con- 
jugate, as  the  rectangles  of  its  abscissas  are  to  the  squares  of  their 
ordinates. 

By  reasoning  similar  to  that  employed  in  Arts.  54. a.,  67.a.,  and 
67.6.,  properties  may  be  inferred,  analogous  to  Props.  B,  E  and  F 
of  the  Ellipse. 


On  the  Properties  of  the  Equilateral  Hyperbola. 

In  Art.  72,  it  was  observed,  that  if  the  axes  of  the  hyperbola  be- 
come equal,  it  is  then  said  to  be  equilateral ;  in  this  case  the  figure 
possesses  some  peculiar  properties,  which  it  may  be  worth  while  to 
investigate. 
C.  s.  11 


82  ON   THE    EQUILATERAL    HYPERBOLA. 

(112.)  Let  the  annexed  figure  represent  an  equilateral  hyper- 
bola, with  its  opposite  and  conjugate  hyperbolas  :  then,  since 
the  axes  ACM,  BCO  are  equal,  it  is  evident  that  if  a  circle  be 


X 


described  upon  the  axis  ACM,  it  will  pass  through  the  extremities 
of  the  axis  BCO,  and  that  the  rectangular  figure  abdc  which  circum- 
scribes those  axes  will  be  a  square.  Draw  the  diagonals  ad,  cb,  and 
produce  them  each  way  to  X,  V,  U,  Z ;  then  XCZ,  UCV  will  be 
the  asymptotes  to  the  four  hyperbolas  ;  and  as  the  angles  aCB,  cCB, 
are  each  of  them  half  a  right  angle,  the  angle  aCc  will  be  a  right 
angle.  Since  the  asymptote  XCV  cuts  the  asymptote  UCV  at  right 
angles  in  the  centre  C,  it  will  also  cut  all  other  lines  BA,  DP,  pQ,, 
&c.  (drawn  parallel  to  UCV)  at  right  angles.  Now  by  Art.  106, 
CRxRP=sA2;   and   for  the  same  reason   CeXeQ*=$A*  ;    .-.  CRx 


ON    THE    EQUILATERAL    HYPERBOLA.  83 

RP=CeXeQ,,  or  CR  :  Ce  : :  eQ, :  RP  ;  hence  if  any  points  R,  e,  &c. 
are  taken  in  the  asymptote,  and  from  them  ordi nates  PR,  eQ,  &c. 
are  drawn  at  right  angles  to  it,  then  the  abscissas  CR,  Ce,  &c.  will 
be  to  each  other  inversely  as  the  ordinates  RP,  eQ,  &c* 

(113.)  Since  the  Latus-rectum  is  a  third  proportional  to  the  major 
and  minor  axes  ;  when  those  axes  are  equal,  it  must  be  equal  to 
either  of  them ;  LST  is  therefore  equal  to  ACM  or  BCO.  Now 
MSxSA=BC2=(since  BC=AC)  AC2 ;  hence  AC  is  a  mean  propor- 
tional between  MS  and  SA  ;  and  since  SYxttz=BC2=AC2,  AC  is  a 
mean  proportional  between  the  perpendiculars  SY  and  Hz. 

(114.)  By  Art.  91.  PC2— CD2=AC2— BC2 ;  but  AC2— BC2=0, 
...  PC2— CD2=0,  consequently  PC=CD,  and  the  diameter  PCG= 
conjugate  DCK.  The  sides  eg,  gh,  hf,  fe  of  the  parallelogram 
eghf,  drawn  about  those  diameters,  will  therefore  be  equal ;  and  the 
parallelogram  itself,  a  Rhombus  whose  area  will  be  equal  to  the 
area  of  the  square  abdc  described  about  the  axes. 

(115.)  Draw  PI  at  right  angles  to  a  tangent  at  P,  and  produce  it 
to  F  ;  then  by  Art.  88,  PIxPF=BC2=AC2 ;  but  CDxPF=ACx 
BC=«AC2 ;  .-.  PIxPF=CDxPF,  and  PI=CD=PC,  i.  e.  the  normal 
PI  is  equal  to  the  distance  PC  from  the  center. 


*  Let   Cs  or  sA  =  a,    CR  =  x,   RP  =  y,  then  (since  CR  x  CP  = 

sA2),  xy=ai,  and  y  = — ,  .-. y#= — '-,  whose  fluent  is  a2xlog.  x-{- 

Cor.  ;  suppose  therefore  the  area  AsRP  to  begin  from  5,  it  would 

vanish  when  a;  =  Cs  or  a ;  hence  a2  x  log.  a  +  Cor.  =  0,  and  Cor. 

=  — a2xlog.  a,  the  area  A^RP  is  therefore  equal  to  a2xlog.  x — 

x 
a2  X  log.  a  =  a2  x  log. -.      Suppose   now   that    Cs=a=l,   then   a8 

and  a  would  each  be  equal  to  1,  and  we  should  have  area  AsRP= 
log.  x ;  and  thus  if  the  abscissas  CR,  Ce,  &c.  are  taken  equal  to  the 
natural  numbers  in  succession,  the  corresponding  areas  AsRP,  AseQ,, 
&c.  will  be  the  Logarithms  of  those  numbers.  It  is  from  this  cir- 
cumstance that  the  system  of  logarithms  whose  modulus  is  unity  are 
called  Hyperbolic  Logarithms. 


84  ON    THE    EQUILATERAL    HYPERBOLA. 

(116.)  Since  ANxNM  :  PN2  ::  AC2  :  BC2,  and  AC2  =  BC2, 
.-.  ANxNM  =  PN2.  Also,  by  Art.  Ill,  FvxvG  :  Q,v2  : :  PC2  : 
CD2;  but  PC2=CD2,  .-.  Vv  x  vG  =  Qv2.  Hence  the  rectangle  of 
the  abscissas  is  equal  to  the  square  of  the  ordinate,  whether  the 
ordinates  be  referred  to  the  axis  or  to  any  diameter  ;  in  this  respect, 
therefore,  the  properties  of  the  equilateral  hyperbola  are  analogous 
to  those  of  the  circle. 

We  have  just  hinted  at  the  analogy  which  obtains  between  the 
properties  of  the  circle  and  those  of  the  equilateral  hyperbola  when 
considered  in  a  geometrical  point  of  view  ;  but  it  appears  more  strik- 
ing when  the  nature  of  those  curves  is  expressed  algebraically.* 
To  pursue  the  inquiry  respecting  this  analogy,  would  lead  to  inves- 
tigations, which,  though  extremely  curious  and  interesting  in  them- 
selves, are  quite  foreign  to  our  present  purposes.  We  therefore  now 
proceed  to  consider  the  nature  of  the  Curvature  of  the  three  Conic 
Sections. 


*  Let  CA=a,  CN=#,  PN=y ;  then,  by  Art.  81.  Cor.  1.  (since 
CA2=BC2  and  .-.CN2— CA2=PN2)  we  have  y2=#2— a\  or  y=* 
V(#2 — a2) ;  now  let  a  be  the  radius  of  a  circle,  x  the  abscissa  meas- 
ured from  the  center  and  y  the  ordinate,  then,  by  the  property  of 
the  circle,  y  =  V(a2 — #2)=v( — l)XV(f — a2);  the  algebraic  ex- 
pression therefore  for  the  ordinate  of  the  circle  is  the  same  with 
the  expression  for  the  ordinate  of  the  equilateral  hyperbola,  except 
as  to  the  imaginary  factor  V( — 1).  This  similarity  in  the  algebraic 
expression  for  the  ordinate,  lays  the  foundation  of  some  very  curious 
analytical  Theorems  with  respect  to  the  analogy  between  these  two 
curves. 


ON  CURVATURE  IN  GENERAL.  85 


CHAPTER  V. 
ON   THE   CURVATURE    OP   THE   CONIC   SECTIONS. 


In  order  to  become  thoroughly  acquainted  with  the  geometry  of 
curvilinear  figures,  it  is  necessary  to  acquire  clear  and  distinct  ideas 
of  the  nature  of  Curvature.  Previous  to  the  investigation,  there- 
fore, of  the  theorems  relating  to  the  curvature  of  the  Conic  Sections, 
it  will  be  very  proper  to  consider  the  nature  of  Curvature  in  General. 

XL 

On  Curvature,  and  the  Variation  of  Curvature. 

(117.)  As  a  straight  line  (AB)  is  defined  to  be  that  which  "lies 

evenly  between  its  extreme  points,"*  A  - — -=s=- B 

so  a  curved  line  (BC)  may  be  said 
to  be  that  which  does  not  "lie  even- 
ly between    those    points  f   and   by  C 
curvature  is  meant  the  continued  deviation  from  that  evenness  of  po- 


*  This  is  the  original  definition  of  Euclid,  and  it  is  retained  by 
Simson,  in  his  edition  of  that  Geometer's  works.  If,  however,  we 
were  left  to  conceive  of  a  straight  line  solely  from  this  definition,  it  is 
questionable  whether  our  conceptions  would  be  very  clear.  "  The 
word  evenly"  as  Playfair  remarks,  " stands  as  much  in  need  of  an 
explanation,  as  the  word  straight,  which  it  is  intended  to  define." 
The  definition  given  by  this  latter  mathematician  is  this.  "  If  two 
lines  are  such,  that  they  cannot  coincide  in  any  two  points  without 
coinciding  altogether,  each  of  them  is  called  a  straight  line."     A 


86 


ON    CURVATURE    IN    GENERAL. 


sition  which  takes  place  in  the  course  of  its  description.  The  curv- 
ature, moreover,  is  said  to  be  greater  or  less,  according  as  that  devia- 
tion is  greater  or  less  within  a  given  distance  of  the  point  from  which 
the  curve  begins  to  be  described.  We  know  not  how  to  illustrate 
this  definition  better,  than  by  referring  the  reader  to  the  annexed  fig- 
ure, where  several  circles  AEM,  AFL,  AGK,  &c.  of  different  di- 
ameters AM,  AL,  AK,  &c.  begin  to  be  described  from  the  point  A, 
all  touching  the  straight  line  BC.  At  the  given  distance  AD  from 
the  point  A,  draw  the  line  DH  at  right  angles  to  AD,  and  cutting 
the  circles  in  the  points  E,  F,  G,  &c.,  then  the  deviations  of  the  cir- 


cles AGK,  AFL,  AEM,  &c.  from  the  right  line  AB,  are  measured 
by  the  lines  DG,  DF,  DE,  &c.  respectively;  and  since  DG  is 
greater  than  DF,  DF  than  DE,  &c.  the  curvature  of  the  circle 


straight  line  being  thus  defined,  the  best  account  that  can  be  given 
of  a  curve  is  to  say,  that  it  is  a  line,  which  cannot  have  a  common 
segment  with  a  straight  line  ;  or  a  line  which  continually  deviates 
from  a  straight  line. 


ON    CURVATURE    IN    GENERAL. 


87 


AGK  is  said  to  be  greater  than  that  of  the  circle  AFL,  of  AFL 
greater  than  that  of  AEM,  &c.  &c.* 

(118.)  Suppose  now  XABCY  to  be  any  curve,  to  which  tangents 
DA,  EB,  FC,  &c.  are  drawn  at  the  points  A,  B,  C,  &c. ;  then, 
from  what  has  been  shown  in  Art.  117,  it  is  evident  that  an  unlimit- 
ed number  of  circles  may  be  described  at  each  of  the  points  A,  B, 


C,  &c.  to  which  the  lines  DA,  EB,  FC,  &c.  shall  be  tangents  as 
well  as  to  the  curve  ;  but  that  there  can  be  only  one  circle,  which 
shall  have  the  same  deviation  from  the  tangent  as  the  curve  at  each 
point.  Let  ALM,  BNO,  CPQ,,  &c.  be  the  circles  which  have  the 
same  deviation  (i.  e.  which  coincide)  with  the  curve  at  the  points  A, 


*  We  have  here  to  observe,  that  although  the  lines  DE,  DF, 
DG,  &c.  are  made  use  of  to  illustrate  what  is  meant  by  greater  or 
lesser  curvature,  yet  the  actual  relation  between  the  curvatures  of 
these  circles  can  only  be  ascertained  by  finding  the  relation  of  DE, 
DF,  DG,  &c  just  at  the  point  of  contact. 


ON    CURVATURE    IN    GENERAL. 


B,  C,  &c. ;  then  these  circles  are  called  the  Circles  of  Curvature  to 
those  points.* 

(119.)  The  change  which  takes  place  in  the  curvature  from  the 
circumstance  of  its  being  measured  at  different  points  A,  B,  C,  &c. 
by  circles  of  different  diameters,  is  called  the  variation  of  curvature 
of  the  curve  ABC. 
~) 

Having  thus  denned  what  is  meant  by  curvature  and  the  variation 
of  curvature,  we  are  next  to  investigate  the  relation  which  takes 
place  between  the  curve  and  the  tangent  just  at  the  point  of  contact. 
This  is  a  subject  of  considerable  difficulty,  inasmuch  as  it  involves 
the  consideration  of  quantities  which  will  not  admit  of  strict  geo- 
metrical comparison,  but  require  a  species  of  minute  analysis,  the 
principles  of  which  are  exhibited  in  the  following  Theorems. 

Theorem  1. 

(120.)  In  the  circle  PQ,VL,  take  any  arc  QP ;  from  P,  Q,  draw 
any  chords  PV,  Q,V,  and  the  tangent  PR  to  the  point  P ;  from  Q, 
draw  Q,R  parallel  to  PV,  and  Qv  parallel  to  RP  ;  join  QJP ;  then,  at 
the  point  of  contact,  the  arc  Q,P,  the  chord  Q,P,  the  tangent  RP,  and 
the  ordinate  Qv,  all  become  equal  to  each  other. 


*  Since  the  curve  and  circle  of  curvature  have  the  same  devia 
tion  from  the  tangent,  at  the  point  of  contact,  it  is  obvious  that  no 
other  circle  can  be  drawn  between.  This  relation  between  the 
curve  and  circle  of  curvature  is  similar  to  that  which  exists  between 
a  circle  and  its  tangent.  Hence  the  circle  of  curvature  is  said  to 
touch  the  curve.  It  will  be  observed,  however,  that  the  circle  often 
cuts  the  curve,  which  it  is  said  to  touch  in  the  point  of  contact. 
This  must  always  be  the  case,  except  at  points  of  maximum  or  min- 
imum curvature,  when  the  circle  falls  wholly  within  or  wholly 
without  the  curve. 


ON  CURVATURE  IN  GENERAL. 


89 


Since  RP  touches  the  circle,  and  PQ  cuts  it,  the  angle  RPQ  is 
equal  to  the  angle  QVP  in  the  alternate  segment ;  and  since  QR  is 
parallel  to  PV,  the  <  RQP=  alternate  <  QPV ;  the  triangles 
PQR,  PQV  therefore  are  similar  ;  hence  we  have  PQ  :  PR  : : 
PV  :  QV.  Now  suppose  the  chord  PV  to  remain  fixed  whilst  the 
chord  QV  revolves  round  the  point  V  by  the  continual  approach  of 
the  point  Q  towards  P,  then  it  is  evident  that  the  chords  PV  and 
QV  continually  approach  towards  a  state  of  equality ;  PQ,  and  PR 
therefore,  which  are  to  each 
other  in  the  ratio  of  PV  : 
QV,  must  also  approach  to 
a  state  of  equality ;  as  must 
also  the  arc  QP  which  lies 
between  PQ,  and  PR,  and  the. 
ordinate  Qv  which  is  equal 
to  PR.  At  the  point  of  con- 
tact, QV  becomes  actually 
equal  to  PV  ;  hence  the  arc 
QP,  chord  QP,  tangent  PR, 
and  ordinate  Qv,  (whose  rela- 
tion is  expressed  by  the  equali- 
ty of  the  determinate  lines  PV 
actually  equal. 


QV)  must  at  that  point  become 


Theorem  2. 


(121.)  The  chord  PV  is  equal  to  ^^- ;  assuming  the  rela- 
tion which  QP  and  QR  have  to  each  other  at  the  point  of  contact. 

By  similar  triangles.  QPR,  PQV,  QR  :  PQ  : :  PQ  :  PV ;  but  (by 

Art.  120.)  at  the  point  of  contact,  chord  PQ=arc  PQ,  .-.  in  this 

Care  QP? 
case  QR  :  arc  PQ  : :  arc  PQ  :  PV ;  hence  PW    aR~ • 

In  order  to  remove  the  objection  which  may  arise  from  the  cir- 
cumstance of  representing  the  definite  quantity  PV  by  the  quantity 
C.  S.  12 


90 


ON    CURVATURE    IN    GENERAL. 


OF 
QR 


,  in  which  QP  and  QR  are  confessedly  too  small  for  geometri- 


cal comparison,  it  should  be  recollected  that  the  measure  of  a  ratio 
is  entirely  independent  of  the  terms  of  a  ratio,  and  consequently  that 
the  two  ratios  which  compose  the  proportion  QR  :  PQ  : :  PQ  :  PV 
are  as  much  real  ratios  at  that  particular  period  when  the  arc  PQ 
may  be  considered  as  equal  to  the  chord  PQ,,  as  at  any  other  period 
of  the  progress  of  the  point  Q  towards  P.  The  conclusion  therefore 
deduced  from  the  reality  of  that  proportion,  viz.  that  PV  is  equal  to 

pa2 


QR' 


must  be  true  in  the  case  when  the  arc  PQ=  the  chord  PQ, 


i.  e.  at  the  point  of  contact. 


Theorem  3. 


(122.)  In  different  circles  the  curvature  varies  inversely  as  the 
radii  of  these  circles. 

Let  AEL,  AFK  be  two  circles  having  a  common  tangent  (BC) 
at  A  :  in  AB  take  any  point  D,  and  draw  DF  at  right  angles  to  AB  ; 
draw  the  chords  AE,  EL  ;    AF,  FK  ;    and  let  fall  Ee,  F/,  perpen- 


E 

<7^ 

fC> 

Sm 

/     \\ 

KX           / 

L.^^/ 

dicular  to  the  diameter  AL  J  then  will  Ae  be  equal  to  DE,  and 
A/  will  be  equal  to  DF.  Now  (by  Euc.  8.  6.)  in  the  right- 
angled  triangles  AEL,  AFK,  we  have  Ae  :  AE  : :  AE  :  AL  ; 

AE8 
.-.  Ae  or  DE=AL-  ;   also  A/  :  AF  : :  AF  :  AK  ;    .-.  A/  or  DF= 


ON  CURVATURE  IN  GENERAL.  91 

AF2  AE2   AF2 

-r^=r :  hence  DE  :  DF  : :  -r^- :  -r-==  .      But    the   curvature   of  the 

AK  AL     AK 

circles  AEL,  AFK,  (see  note  page  88,)  is  measured  by  the  relation 
which  obtains  between  DE  and  DF  just  at  the  point  of  contact ;  and 
at  that  point,  AE  and  AF  both  become  equal  to  AD  (by  Art.  120.) 
and  consequently  equal  to  each  other.  At  the  point  of  contact,  there- 
fore,  (since   AE2=AF2)  we    have   DE  :  DF  :  :  J*   :  -?-  : :  AK  : 

AL    AK 

AL ;  i.  e.  curvature  of  circle  AEL  :  curvature  of  circle  AFK  : :  di- 
ameter of  AFK  :  diameter  of  AEL  : :  radius  of  AFK  :  radius  of 
AEL  ;  i.  e.  the  curvature  in  different  circles  varies  inversely  as  their 
radii. 


Theorem  4. 

(123.)  Let  now  APQ,  be  any  curve,  PVO  the  circle  op  cur- 
vature to  the  point  P ;  take  any  arc  PCI  and  through  Q,  draw  RQ,q 
parallel  to  the  chord  P V  passing  through  some  given  point  S  ;  then 
(assuming  the  relation  of  the  quantities  PQ,  and  Q,R  at  the  point  of 

PQ,2 


contact)  PV  will  be  equal  to  :=-=-, 


P<72 
By  Theorem  2,  PV  is  equal  to  -*- ;  but  since  the  curve  and 

qix 

circle  of  curvature  coincide  at  the  point  of  contact,  at  that  point 


92 


ON  THE  CURVATURE  OF  THE  PARABOLA. 


P^  will  become  equal  to  PQ,,  and  qR  equal  to  Q,R,  and  consequently 

pa8 


PV= 


qr 


(124.)  Draw  now  VO  at  right  angles  to  PV,  and  join  PO ;  then 
(PVO  being  a  right  angle,  consequently  in  a  semi-circle)  PO  will 
be  the  diameter  of  curvature  to  the  point  P.  Bisect  PO  in  r,  then 
Vr  will  be  the  radius,  and  r  the  center  of  curvature  to  the  point  P. 


XII. 


On  the  Curvature  of  the  Parabola. 

Let  AQ,P  be  a  Parabola,  whose  axis  is  AZ,  and  focus  S ;  and  let 
PVO  be  the  circle  of  a  curvature  to  any  point  P.  Join  SP,  and 
produce  it  to  meet  the  circle  of  curvature  in  V,  then  PV  is  the  chord 
of  curvature  passing  through  the  focus. 


(125.)  The  Chord  PV=4SP.     Take  any  arc  QP,  so  small  that 
it  may  be  considered  as  coinciding  with  the  circle  of  curvature,  and 


ON  THE  CURVATURE  OF  THE  PARABOLA.  93 

draw  Q,R  parallel  to  SP  ;  draw  also  Qiv  parallel  to  the  tangent  PT, 
cutting  SP  in  x,  and  the  diameter  PW  in  v  ;  then  Q,RP#  will  be  a 
parallelogram,  and  P#  will  be  equal  to  Q,R.  Now  since  xv  is  par- 
allel to  PT,  and  Vv  parallel  to  TS,  the  &Vxv  is  similar  to  the  APST  ; 
but  by  Art.  17,  SP  is  equal  to  ST  ;  .-.  P#=Pv  ;  hence  Vv  is  equal 
to  Q,R.  Let  Qiv  move  up  towards  P  parallel  to  itself,  then,  at  the 
point  of  contact,  Qiv  will  become  equal  to  Q,P  ;*  since  therefore  Vv= 

(Q  P2  v  Q  v2 

%s-  by  Art.  123.  =  )^~.      But 
U,K  /  Pv 

by  Art.  22,  4SPxPv=Q,i;2 ;  .-.  ^=4SP  ;  hence  PY=4SP=  pa- 
rameter to  the  point  P. 

That  is,  the  chord  of  curvature,  passing  through  the  focus,  is 
equal  to  the  parameter  of  the  diameter  at  the  point  of  contact. 

(126.)  SA.P02=16SP5.  Draw  YO  at  right  angles  to  PV  and 
join  PO.  Then  (124.)  PO  is  the  diameter  of  curvature,  and  there- 
fore parallel  to  SY,  which  is  perpendicular  to  the  tangent  PT. 
Hence  the  triangles  PYO,  SYP  are  similar 

.-.  PO   :  PY(=4SP)  : :  SP  :  SY, 
PO2  :      16SP2      : :  SP2  :  SY2(=SA.SP,  by  Art.  32.) 
PO2:      16SP2      ::SP   :  SA, 
.-.SA.P02=16SP3. 

Therefore,  a  parallelopiped,  whose  base  is  the  square  of  the  diam- 
eter of  curvature,  and  whose  height  is  the  distance  from  the  focus 
to  the  vertex,  is  equal  to  16  times  the  cube  of  the  dist;mce  from  the 
focus  to  the  point  of  contact. 

4gpf 
Cor.  1.   The  diameter  of  curvature  ■»  ;>^>v. 

V(SA) 


#  By  Art.  120,  Qx  becomes  equal  to  QP  ;  but  at  the  point  of  con- 
tact P,  the  points  x  and  v  coincide ;  therefore  at  that  point  the  three 
lines  QP,  Q#,  Qiv  become  equal  to  each  other. 


94  ON   THE    CURVATURE    OF    THE    ELLIPSE. 

PV3  pvi 

Cor.  2.  SA.P02=-.~  and  PO=£r7^r-rV 

4  2V(SA)     , 

Cor.  3.  The  diameter  PO  (and  of  course  the  radius  Pr)  ocSP* 

3. 

or  PV2,  because  SA  is  constant. 

(127.)  At  the  vertex  A,  where  SP  becomes  perpendicular  to  the 
tangent,  the  chord  and  diameter  of  curvature  will  of  course  coincide  ; 
and  in  this  case  each  of  them  becomes  equal  to  4SA,  i.  e.  to  the  la- 
tus-rectum.*  The  diameter  (and  consequently  the  radius)  of  cur- 
vature is  therefore  the  least  at  A  ;  hence,  by  Art.  122,  the  curvature 

itself  will  be  greatest  at  A  ;  and  since  it  varies  as  =-,  i.  e. ,  it 

will  keep  continually  decreasing  as  the  point  P  recedes  from  A. 


XIII. 

Chi  the  Curvature  of  the  Ellipse.     (Fig.  in  next  page.) 

Let  APM  be  an  ellipse,  PVLO  the  circle  of  curvature  to  the  point 
P  ;  join  PS,  PC,  and  produce  them  to  meet  the  circle  of  curvature 
in  the  points  V,  L ;  draw  YO,  LO  at  right  angles  to  PV,  PL,  and 
join  PO  ;  then  PV  is  the  chord  of  curvature  passing  through  the  fo- 
cus ;  PL  the  chord  passing  through  the  center  ;  and  PO  the  diame- 
ter of  curvature.     Draw  the  conjugate  diameter  DCK  ;  then, 

(128.)  The  chord  of  curvature  (PL)  passing  through  the  center 

2CD2 
is  equal  to  -«*• .     Take  any  small  arc  Q,P  as  before  ;  draw  Q,R 

parallel  to  PC,  and  Q,v  parallel  to  RP ;  then  will  Pv  be  equal  to 
RQ,.     Suppose  Qv  to  move  up  towards  P,  then,  at  the  point  of  con- 


*  For  at  A,  SP  becomes  equal  to  SA  ;    .-.  PV=»4SP=~4SA  ;    and 
„>     4SP*       4SA*      ... 


ON   THE    CURVATURE    OP   THE    ELLIPSE. 


95 


tact,  Qv  becomes  equal  to  Q,P  (120.),  and  vG  becomes  equal  to 
PG,  i.  e.  to  2PC.  Now  by  Art.  53,  FvxvG  :  Qv2  : :  PC2  :  CD2  ; 
substituting  therefore  for  Pv,  vG  and  Q,v,  their  values  at  the  point  of 

contact,  we  have  QRx2PC  :  GIP2  : :  PC2  :  CD2,  or  2PC  :^~  :: 


2CD5 


123.),  .•.PL=-p^-. 


(129.)  The  diameter  of  curvature  (PO)= 


2CDS 


PF 


The  triangles 


PCF,  PLO  have  a  common  <  at  P,  and  right  <s  at  L  and  F, 


2CDS 


-p-Q- J  : :  PC  :  PF, 

2CD2xPC     2CD8      .         L        -  /T>  *      .„ 

;  the  radius  of  curvature  (Pr)  will  con- 


•.PO= 


PCxPF 


PF 

sequently  be  equal  to  „-. 


96  ON   THE    CURVATURE    OP    THE    ELLIPSE. 


(129.a.)  The  chord  of  curvature  (PY)  passing  through  the  focus 

U  ™—.     The  triangles  PEF,  PVO,  have  a  common  <  at  P  and 

AC 

right  <s  at  V  and  F,  they  are  therefore  similar.     Hence  PV  :  PO 

(--)i:PP:PE(iC),,PV=™-^.    A,B„H, 

extremity  of  the  minor  axis)  the  semi-conjugate  becomes  equal 
to  AC  ;  hence  the  chord  of  curvature  passing  from  B  through  the 

2  AC2 

focus  S=^=2AC. 

(130.)  At  A  (the  extremity  of  the  major  axis)  the  diameter  of 

(2CD2,v      2RC2 
___._  j=___r=  latus-rectum  ;*    at  B  (the  extremity 

2AC2 

of  the  minor  axis)  it  is  equal  to  -^^  ;   it  is  therefore  least  at  A, 

and  greatest  at  B  ;  hence,  by  Art.  122,  the  curvature  is  greatest  at 
the  extremity  of  the  major  axis,  and  least  at  the  extremity  of  the 
minor  axis.  At  the  intermediate  points  between  the  extremities  of 
the  axes,  the  curvature  varies  inversely  as  the  cube  of  the  normal.! 


rectum,  i.  e.  2AC  :  2BC  : :  2BC  :  latus-rectum 


*  For  by  Art.  39,   major  axis  :  minor  axis  : :  minor  axis  :  latus- 

4BC8=2BC2 
2AC       AC* 

CD8 

t  The  radius  of  curvature  ==^^ .     Now  by  Art.  58,  (PI  being  the 

Pr 

normal)    PIxPF=BC2=  a  constant  quantity,    .-.Plac^^.      By 
Art.  62.   CDxPF=ACxCB=  a  constant  quantity,    .-.CDoc^; 

Pr 
1       CD2/  1  \ 

hence  PI  acCD.  Again,  since  CD  oc™  =  li.  e.  CD2XppJ 
ocCD3  ocPI3  ;  the  radius  of  curvature  therefore  varies  as  PIS,  con- 
sequently the  curvature  itself  varies  as  p=-,  or  inversely  as  the  cube. 


ON  THE  CURVATURE  OP  THE  HYPERBOLA. 


97 


XIY. 

On  the  Curvature  of  the  Hyperbola. 

The  process  for  rinding  the  chords  and  diameter  of  curvature  in 
the  Hyperbola  is  precisely  the  same  as  that  for  the  Ellipse.  Refer- 
ring the  reader  to  the  annexed  Figure,  we  shall  merely  repeat  the 
principal  steps  of  the  foregoing  demonstration. 


(131.)   *By  Art.   111.,   Vv  x  vG  :  Qv2  : :  PC2  :  CD2,  and  at  the 
point'  of  contact,  QJEtx2PC  :  QP2  : :  PC2  :  CD2,  .-.S§-  or  PL- 


of  the  normal.     As  PI  ocCD,  the  curvature  varies  as  p^pr--  gc^^— 

OU3     Dn.3, 

or  inversely  as  the  cube  of  the  diameter  conjugate  to  that  at  the 

point  of  contact. 


*  The  construction  of  the  above  figure  is  word  for  word  the  same 
as  in  the  Ellipse.  To  avoid  a  confusion  of  lines,  the  circle  of  cur- 
vature is  drawn  entirely  within  the  Hyperbola ;  whereas,  such  part  of 
the  hyperbola  as  is  of  greater  curvature  than  that  at  the  point  P, 
ought  to  have  fallen  within  the  circle  of  curvature,  as  in  Figure, 
page  87. 

c.  s.  13 


98         ON  THE  CURVATURE  OF  THE  HYPERBOLA. 

2CD2 


chord  of  curvature  passing  through  the  center. 


PC 

(2CD2>v 
"per)  :: 

2CD2  CD2 

PC  :  PF,  .-.  PO=— «^-  ;  and  Pr  the  radius  of  curvature  =^=. 

-pp-  )  ::  PF 

SCD 

"AC 


2CD2 

:  PE(AC),  .-.  PV=  --^-=  chord  of  curvature  passing  through  the 


focus. 

r  2PT)2      2RC2 

(134.)  At  the  vertex  A,  the  diameter  of  curvature  — -  «■  — r-^ 

=  latus-rectum.  Here  the  analogy  between  the  Ellipse  and  the 
Hyperbola  ends  ;  for  with  respect  to  the  variation  of  curvature,  since 
the  normal  PI  keeps  continually  increasing  from  the  point  A,*  the 
curvature  will  continually  decrease  as  the  point  P  recedes  from  A. 

(135.)  In  the  equilateral  hyperbola  (see  Fig.  in  page  82)  the  latus- 
rectum  is  equal  to  the  major  axis  ;  the  curvature  therefore  at  the 
vertex  A  is  the  same  with  the  curvature  of  the  circle  described  upon 
the  major  axis.  In  this  case  PI=PC  (Art.  115);  .-.PI3ocPC3, 
and  in  the  recess  of  P  from  the  point  A.  the  curvature  varies  in  the 


same 


ratio,  viz.  (      -  or  ^pr3  )  with  respect  to  the  two  sides  of  the 

isosceles  triangle  CPI,  one  of  which  (PC)  revolves  round  the  fixed 
point  C,  and  the  other  (PI)  round  the  moveable  point  I,  at  right  an- 
gles to  the  curve.  Here  then  is  an  instance  of  great  symmetry  in 
the  curvature  of  the  equilateral  hyperbola. 


*  That  the  radius  of  curvature  varies  as  the  cube  of  the  normal, 
is  proved  in  the  same  manner  as  in  Note  f,  page  96. 


ANALOGOUS  PROPERTIES,  &C.  99 


CHAPTER  VI. 

ON  THE  ANALOGOUS  PROPERTIES  OF  THE  THREE  CONIC 

SECTIONS. 


Hitherto  we  have  noticed  no  other  analogies  than  those  which 
take  place  between  the  Ellipse  and  Hyperbola ;  but  as  the  three 
Conic  Sections  are  derived  from  the  same  solid  merely  by  changing 
the  position  of  the  plane  which  intersects  its  surface,  it  may  naturally 
be  expected  that  they  will  possess  many  properties  common  to  them 
all.  Previous  to  the  investigation  of  these  analogous  properties,  it 
may  be  worth  while  to  consider  the  changes  which  take  place  in  the 
nature  of  the  section,  during  the  revolution  of  the  plane  of  intersec- 
tion from  a  position  parallel  to  the  base  of  the  cone,  till  it  becomes  a 
tangent  to  one  of  its  sides. 

XV. 

On  the  changes  which  take  place  in  the  nature  of  the  curve  describ- 
ed upon  the  surface  of  a  cone,  during  the  revolution  of  the  plane 
of  intersection. 

(136.)  Let  the  triangle  BEZG  represent  the  section  of  a  cone 
perpendicular  to  its  base,  and  passing  through  the  vertex ;  then  if  the 
cone  be  cut  by  a  plane  perpendicular  to  BEZG,  and  parallel  to  the 
base,  the  section  AFD  will  be  a  circle.  Draw  the  diameter  AD  of 
the  circle  AFD,  and  draw  AZ  parallel  to  the  side  BE  of  the  cone. 
Conceive  a  plane  (at  right  angles  to  the  plane  BEZG)  to  pass 
through  AD,  and  afterwards  to  revolve  through  the  angle  DAG  till 
it  becomes  a  tangent  to  the  side  BG  of  the  cone.  From  what  was 
shown  in  Chapter  I.  it  is  evident  that  whilst  this  plane  revolves 


100 


ANALOGOUS    PROPERTIES    OF 


through  the  angle  DAZ,  its  intersection  APM  with  the  surface  of 
the  cone  will  be  an  Ellipse,  whose  major  axis  is  AM ;  when  it  comes 


into  the  position  AZ,  it  will  be  a  Parabola,  whose  axis  is  AZ  ;  and 
that  whilst  it  revolves  through  the  angle  ZAG,  it  will  be  an  Hyper- 
bola, whose  major  axis  is  AM',  M'  being  the  intersection  of  z A  and 
EB  produced. 

It  may  further  be  observed,  that  in  the  revolution  of  the  plane 
through  the  angle  DAZ,  so  long  as  it  cuts  the  side  BE  between  D, 
and  E,  a  whole  ellipse  will  be  formed  upon  the  surface  of  the  cone. 
When  it  comes  into  such  a  position  as  to  cut  the  base,  a  part  only  of 
an  ellipse  will  be  formed ;  and  when  it  arrives  at  the  position  AZ, 
the  point  M  moves  off  to  an  infinite  distance,  so  that  the  Parabola 
thus  formed  may  be  considered  as  a  part  of  an  Ellipse,  whose  axis 
major  is  infinite.  And  as  at  the  instant  the  plane  leaves  the  position 
AZ  in  direction  Zz,  the  curve  of  intersection  becomes  an  Hyperbola, 


THE    THREE    CONIC    SECTIONS.  101 

the  Parabola  may  also  be  regarded  as  an  Hyperbola,  whose  major 
axis  is  infinite.  These  three  curves  therefore  approach  to  identity 
at  the  same  time  that  the  plane  approaches  to  parallelism  with  the 
side  BE  of  the  cone. 

(137.)  The  same  conclusion  may  be  drawn  from  the  algebraic 
construction  of  these  curves.  Let  the  angle  MAZ  be  equal  to  the 
angle  ZAz,  then  the  major  axis  (AM')  of  the  Hyperbola  will  be 
equal  to  the  major  axis  (AM)  of  the  Ellipse.*  In  each  case,  find 
the  center  C  or  C,  and  let  the  abscissas  AN  or  AN'  =#,  the  ordin- 
ate PN  or  P'N'=y,  semi-axis  major  (AC  or  AC')=a,  semi-axis 
minor  =&,  AS  or  AS'  (S  or  S'  being  focus)  =c.  Then  in  the 
Ellipse  NM=AM— AN=2a— x,  and  MS=AM— AS-2a— c  ;  in 
the  Hyperbola,  N'M'=AM'+AN'==2a+^,  and  M'S'=AM'+AS'= 
2a-fc.  Now  by  Arts.  46,  81.  (see  Figs,  in  pp.  40,  64.)  we  have 
ANxNM  or  AN'xN'M'  :  PN2  or  P'N'2  : :  AC2  :  BC2,  or  xx{2a±x) 
:  y2  : :  a%  ;  62. 

V- 
Hence  y2=-j x(2ax±x*)  is  the  general  equation  between  the  ab- 
a 

scissa  and  ordinate  of  the  ellipse  and  hyperbola. 

But  in   the   Ellipse    MSxSA  =  BC2,    and  in   the    Hyperbola 

M'S'xS'A  =  BC2,   or   (2a±c)  Xc=l»2,   hence  by  substitution  y2= 

2ac4-cz     "L  "  .  2c*x     2cx2    ,    cV       _         .  * 

^—  x(2ax  +  x2)4.cx4-  = 4- h  — w  •     Conceive  now  the 

a2         v  '  a  a  a* 

angles  MAZ,  ZA#  to  be  continually  diminished,  then  the  axis  major 

both  in  the  Ellipse  and  Hyperbola  is  continually  increased,  and  just 

at  the  instant  of  their  approach  to  coincidence  with  the  line  AZ,  each 

of  them  becomes  indefinitely  great ;  in  which  case,  (supposing  x  and 

c  to  be  finite  quantities)  the  three  fractional  terms  of  the  last  equa- 


*  Since  AZ  is  parallel  to  M'BE,  the  angles  MAZ,  ZAz  are  re- 
spectively equal  to  the  angles  of  the  triangle  MAM' ;  which  triangle 
is  therefore  isosceles. 


102  ANALOGOUS    PROPERTIES    OF 

tion  become  equal  to  nothing;  .-.  y2=4cr,  or  PN2=4ASxAN, 
which  is  the  property  of  the  Parabola.  Hence  it  appears  that  a  fi- 
nite part  of  an  Ellipse  or  Hyperbola  whose  latus-rectum  is  finite,  but 
whose  axis  major  is  infinite,  may  be  considered  as  a  Parabola ;  and 
vice  versa,  that  a  finite  part  of  a  parabola  may  be  considered  as  a 
part  of  an  ellipse  or  hyperbola,  whose  axis  major  is  infinite,  and 
latus-rectum  finite. 

XVI. 

On  the  mode  of  constructing  the  Three  Conic  Sections  by  means  of 
a  Directrix,  and  the  Properties  derived  therefrom. 

In  Chap.  I.  we  have  already  shown  the  method  of  constructing  the 
Parabola  by  means  of  a  directrix  ;  we  now  proceed  to  show  that  the 
Ellipse  and  Hyperbola  may  also  be  constructed  by  lines  revolving 
in  a  similar  manner. 

(138.)  Let  MED  be  a  line  given  in  position  ;  and  from  the  point 
E,  draw  CEC  at  right  <  s  to  MED ;  in  CEC  take  any  point  A, 
and  set  off  AS :  AE  : :  m  :  1.  Let  the  line  SP  begin  to  revolve 
from  A  round  S,  and  PM  move  parallel  to  EC,  in  such  manner  that 
SP  may  be  always  to  PM  as  AS  to  AE  (i.  e.  in  the  given  ratio  of 
m  :  1.) ;  then  the  curve  generated  by  the  point  of  intersection  P  will 
be  one  of  the  Conic  Sections. 

Let  fall  PN  at  right  <  s  to  AC,  and  let  AN=#,  PN=y,  AS=c ; 

then,  since  AS  (c)  :  AE  : :  m  :  1,  we  have  AE=—  ;     now     PM  = 

m  ' 

NE=AE  +  AN=— +ar,  and  SP  :  Pm(— +A  :  :  m  :  1  ;   ...  SP= 
m  \m      / 

c+mx  ;  also  SN=AN— AS=:r— c. 

Hence  we  have,  SP2  =  (c+ra:r)2=c2-f2cra.r+raV2, 

SN2  =  (x—cf  =  c*—2cx+x* ; 

.7.  SP2— SN2(=PN2)-y2=(w-f-l)2c2:-l-(m2— l).r«r 


ON   THE    THREE    CONIC    SECTIONS. 


103 


(139.)  Let  ra=l,  or  SP  =  PM,  then  m  +  l=2,  and  m2-— 1  =  0, 
...  y2=4c#,  or  PN2=4ASxAN;  hence  ALP  is  a  parabola,  whose 
vertex  is  A,  focus  S,  and  axis  AC. 


B 

M 

•'B 

C 

/ 

P 

E 
D 

a!   s 

T 

N        c 

(140.)  Let  m  be  less  than  1,  or  SP  less  than  PM.  On  the  same 
side  of  A  with  PN,  take  AC  :  SC  : :  1  :  m,  or  AC  :  AC— SC(= 
SA=c)  : :  1  :  1 — m,  then  c=(l — m).AC  ;  hence  (m  +  \)2cx  =  (m -f 
1)  (l—m)x2AC.x=(l—m2).2AC.a;.  From  C  draw  BC  at  right 
angles  to  AC,  and  take  BC2  :  AC2  : :  1— m2  :  1,  then  1— m%  = 
BC2 


B°2  A         .       1 

—  2,   and  m2— 1= 


AC2' 


Substitute  these  values  for  1 — m2  and 


BC5 


m2  — 1,  and  we  have  (m  +  l)2c#=T7^x2AC.#,  and  (m2 — \)x2= 


AC5 


BC 


""AC2*^5  nowlet  AC==«>  BC=6,  then  y2=((ra-hl)2c#-f(ra2-^ 

l)^2=)^x(2aa:— x*).     Hence  by  (Art.   137)  ALP  is  an  ellipse, 
whose  semi-axis  major  =*AC,  semi-axis  minor  =BC,  and  focus  S.* 


*  To  prove  that  S  is  the  focus,  we  have  AC  :  SC  : :  1  :  m,  .-. 
AC2  :  SC2  : :  1  :  m\  and  AC2  :  AC2— SC2  : :  1  :  1— m*  ;  but 
AC2  :  BC2  ::  1  :  1—  m\  .-.  BC2  =  AC2— SC2,  and  SC2  =  AC2— 
BC2. 


104  ANALOGOUS    PROPERTIES    OF 

(141.)  Let  m  be  greater  than  1,  or  SP  greater  than  PM.     Take 

C  on  the  other  side  of  A  in  such  a  manner  that  AC  :  SC  : :  1  :  m, 

or  AC  :  SC— AC(=AS=c)  : :  1  :  m— 1,   then   c=(wfr—  1).AC,  and 

(m  +  l)2c=(m-f  l)(m— l).2AC  =  (ra2— 1).2AC.      From  C   draw 

BC  at  right  <  s  to  AC,  and  take  BC2  :  AC2  : :  m2— 1  :  1,   then 

RC2 
m2 — I  —  xt^'     ^et  BC=6,  AC=a,  and  substituting  as  before,  we 

b2 
have  y*——(2axJra;2)]  hence  ALP  is  an  hyperbola,  whose  semi- 

axis  major  is  AC,  semi-axis  minor  BC,  and  focus  S. 

Prom  this  mode  of  describing  the  three  Conic  Sections  we  deduce 
the  following  properties. 


Property  1. 

If  a  tangent  be  drawn  to  the  extremity  of  the  latus-rectum  of  any 
conic  section,  it  will  cut  the  axis,  or  the  axis  produced,  in  the  same 
point  with  the  directrix. 

Draw  the  latus-rectum  LST ;  let  LE  be  a  tangent  to  the  curve 
at  L,  and  cut  the  axis  in  E  ;  then 

(142.)  In  the  Parabola,  SP  =  PM,  .-.  AS  =  AE;  hence  SE  = 
2 AS.  But  by  Art.  18,  the  sub-tangent  SE=  twice  the  abscissa  AS, 
.-.  E  is  the  extremity  of  the  sub-tangent,  and  also  a  point  in  the  di- 
rectrix. 

(143.)  In  the  Ellipse,  AS  :  AE  : :  m  :  1,  m  being  less  than  1. 

By  construction  (140)  SC  :  AC  : :  m  :  1. 

.-.  (Euc.  12.  5.)  SC  :  AC  : :  SC-fAS(AC)  ;  AC+AE(EC). 

Therefore  EC  is  a  third  proportional  to  SC  and  AC ;  which  is 
also  true  (55.)  if  E  be  the  point  where  the  tangent  cuts  the  axis  pro- 
duced.    Hence  E  is  a  point  both  in  the  directrix  and  tangent. 

(144.)  In  the  Hyperbola, 

AS  :  AE  : :  m  :  1,  m  being  greater  than  1. 


THE    THREE    CONIC    SECTIONS. 


105 


By  construction  (141.)  SC  :  AC  : :  m  :  1 ; 

.-.  (Euc.  19.  5.)  SC  :  AC  : :  SC— AS(AC)  :  AC— AE(EC). 

*  Therefore,  EC  is  a  third  proportional  to  SC  and  AC ;  which  is 
also  true  (85.)  if  E  be  the  point  where  the  tangent  cuts  the  axis. 
Hence,  E  is  a  point  both  in  the  directrix  and  tangent. 

(145.)  This  line  LE,  which  is  drawn  touching  the  curve  at  the 
extremity  of  the  latus-reclum,  is  called  the  focal  tangent ;  from  what 
has  just  now  been  proved,  it  follows  therefore  that  if  a  line  be  drawn 
at  right  angles  to  the  axis  from  the  point  where  it  is  intersected  by 
the  focal  tangent,  that  line  will  be  the  directrix.* 

Property  A. 

(145.a.)  In  the  cone  YYZ,  let  APQ  be  any  conic  section,  and 
BHG  an  inscribed  sphere,  touching  the  cone  in  the  circle  BDG,  and 


the  plane  of  the  conic  section  APQ,  in  S.     Then  S  is  the  focus  of 
the  conic  section  APQ,.     Also,  if  the  plane  of  the  circle  BDG  be 


*  The  substance  of  Arts.  138  to  145,  inclusive,  may  very  readily 
be  inferred,  without  the  aid  of  Algebra,  from  Arts.  18.a.,  57.a.,  and 

C.  S  14 


106  ANALOGOUS    PROPERTIES    OF 

produced  to  intersect  the  plane  of  the  conic  section  APQ  in  EF, 
then  EP  is  the  directrix  of  the  conic  section  APQ,. 

Let  YYZ  be  a  plane  passing  through  the  axis  VC  of  the  cone,  cut- 
ting the  plane  of  the  conic  section  APQ,  perpendicularly  in  AW,  the 
axis  of  the  conic  section  (1,  3  and  5,)  and  cutting  the  circle  BDG 
in  the  line  BG.  Since  YB  and  VG  are  tangents  to  the  sphere  from 
the  same  point  Y,  they  are  equal*  and  the  axis  VC  of  the  cone,  which 
bisects  the  angle  BVG,  cuts  BG  at  right  angles.  For  the  same  rea- 
son, the  axis  YC  cuts  all  other  lines  passing  through  K  in  the  plane 
of  the  circle  BDG  at  right  angles,  and  this  plane  is,  therefore,  per- 
pendicular to  the  axis  YC,  and  consequently,  to  the  plane  YYZ, 
which  passes  through  it.  Since,  therefore,  the  planes  of  the  circle 
BDG,  and  the  conic  section  APQ  are  both  perpendicular  to  the 
plane  YYZ,  their  common  intersection  EF  is  perpendicular  to 
VYZ,  and  therefore  to  the  lines  BX,  WX,  which  it  meets  in  that 
plane. 

Draw  YL,  in  the  plane  YYZ,  parallel  to  AW,  intersecting  GB, 
produced  if  necessary,  in  L.  From  any  point  P  in  the  curve  APQ, 
draw  PM  at  right  angles  to  EF.  PM  is  parallel  to  WX,  and  con- 
sequently to  VL.  Join  VP,  intersecting  the  circumference  of  the 
circle  BDG  in  D.     Join  LD,  DM. 

Since  D  is  in  the  plane  of  the  parallels  PM,  VL,  the  lines  LD, 
DM  are  in  that  plane.  But  they  are  also  in  the  plane  of  their  circle 
BDG.  Therefore  they  are  in  the  common  intersection  of  the  two 
planes,  and  are  in  the  same  straight  line.  Now  VD=VB,  because 
both  are  tangents  to  the  sphere  from  the  same  point  Y.  For  the 
same  reason  PS=PD. 

And  (sim.  tri.)  PD  :  PM  : :  YD  :  YL, 

PS  :  PM  : :  VB  :  YL,  a  constant  ratio. 


*  For  the  plane  of  the  lines  YB  VG  cuts  the  sphere  in  a  circle, 
to  which  YB  and  YG  are  tangents.  Hence  it  follows  from  Euc.  36. 
3,  that  VB=YG. 


THE    THREE    CONIC    SECTIONS. 


107 


Hence,  the  distance  SP,  of  any  point  of  the  curve  P,  from  S,  is  in 
a  constant  ratio  to  the  perpendicular  PM,  to  the  line  EF  ;  which  is 
the  property  of  the  focus  and  directrix  of  the  conic  section  APQ,. 
Therefore  S  is  the  focus,  and  EF  the  directrix. 


Property  2. 

In  any  Conic  Section,  the  distance  SP= 
half  latus-rectum 

on  • 

1— ~,cos.  <PSN 

AC 

(146.)  Let  radius  =  1,  then  SP  :  SN  : :  1  :  cos.  <  PSN ;  there- 
fore SN=SPxcos.  PSN.  Now,  in  the  Ellipse  and  Hyperbola, 
SP  :  PM  : :  m  :  1,   and  SC  :  AC  : :  m  :  1 ;    .-.  SP  :  PM(=NE= 

SE+SN)  ::  SC  :  AC;    hence  SPxAC  =  SExSC +SCxSN= 


I) 

. 

B 

c 

L.  / 

P 

r, 

E 
D 

aI    s 

T 

I 

¥ 

BC»  +  SC  x  SN  =  BO  +  SC  x  SP  x  cos.  PSN  ;    or  SP  x  AC— 
SPXSCXCOS.PSN-BO.;    therefore  SP-^^-^^-^ 


108  ANALOGOUS    PROPERTIES    OF 

BC2  1  half  latus-rectum* 


x- 


AC     i_^.cos.PSN    1-^' cos.  PSN 
AO  AO 

(147.)  In  the  Parabola,  SC  may  be  considered  as  equal  to  AC  ;t 

_,_      half  lat.  rect.      rnu  .  .  ,      .  ,   , 

.-.  SP  =  ■: ==5,     The  same  expression  might  also  be  dedu- 

1 — cos.  PSN 

ced  immediately  from  the  properties  of  the  Parabola,  for  since  SE— 

2AS,  SP(=PM=NE==SE+SN)=2AS+SN=2AS  +  SPxcos.  PSN, 

*n     m  ticitvt    oao       j  on  ^A^  half  lat.  rect. 

.-.SP— SPxcos.  PSN=2AS,  andSP=- sa^—i S5i5- 

1 — cos.  PSN    1 — cos.  PSN 

(148.)  By  means  of  this  property  we  are  enabled  to  find  the  va- 
riation of  the  distance  SP  in  its  angular  motion  round  the  focus  S ; 
and  in  this  respect  it  forms  an  important  theorem  in  Physical  Astron- 
omy. To  put  the  expression  just  now  deduced  into  the  Algebraic 
form  adopted  by  Mr.  Vince  (at  page  26  of  his  '  Physical  Astronomy') 
in  tracing  the  radius  vector  (SP)  round  the  elliptic  orbit  of  the  moon, 
let  AC=1,  BC=c,  SC=w,  <  PSN=*;  then 


*When   P  is   at  L,    <PSN=a  right  angle;  .-.cos.  PSN=0, 

and  SP=£  latus-rectum.     When  P  is  between  L  and  A,  cos.  PSN 

__       \  latus-rectum 
is  negative  ;  .-.  SP= «_■' ' 

1+AOXCOS,PSN 

t  For  by  Art.  137,  the  Parabola  may  be  considered  as  an  Ellipse, 
whose  major  axis  is  infinite ;  in  this  case  C  goes  off  to  an  infinite 
distance,  and  the  difference  (AS)  between  AC  and  SC  vanishes  with 
respect  to  the  quantities  themselves,  which  may  therefore  be  assum- 
ed as  in  a  ratio  of  equality. 


THE    THREE    CONIC    SECTIONS. 


109 


SP 


< 


BC* * 

AC— SCxcos.  PSN 
1 


w 


WXCOS.  2? 


=  CaX: 


=(by  actual  division) 


1 — wxcos.  z 

c2x(l  +  wXcos.  ^+2^2x(cos.  z)2+t03x(cos.  z)3+,  (fee.) 
For  the  trigonometrical  transformation  of  this  expression,  and  its 
practical  application,  we  refer  the  reader  to  the  work  itself. 

(149.)  Before  we  leave  this  subject  of  the  radius  vector,  it  may 
not  be  improper  to  show  its  variation  with  respect  to  an  angle  de- 
scribed about  the  center  of  the  Ellipse.  Upon  the  major-axis  AM 
describe  the  semi-circle  AQ,M,  produce  NP  to  Q,,  and  join  Q,C. 

Q    R 


Draw  PH  to  the  other  focus,  then  PN2=SP2— SN2=HP2— HN2 ; 
.-.  HP2— SP2=HN2— SN2. 

Hence  we  have, 

(HP-fSP).(HP— SP)=(HN+SN).(HN— SN) ;  - 


HN— SN  :  HP— SP, 
2CN     :2AC— 2SP; 
CN     :  AC— SP, 
AC(orQ,C):CN 
1 :  cos.  aCN ; 
.-.SCxcos.  aCN  =  AC— SP, 


.-.  HP+SP  :  HN-f-SN 
i.e.  2AC:HSor2SC 
or  AC  :       SC 

.-.so:  ac-sp 


or  SP=  AC — SC  xcos.  Q,CN,  which  is  an  expression 
for  the  radius  vector,  with  reference  to  the  center. 


Property  3. 

(150.)  If  a  conic  section  be  cut  through  the  focus  (S)  by  a  line 
(Vp)  terminated  at  each  extremity  by  the  curve,  then  4SPxSp= 
latus-rectum  x  P/>. 


110 


ANALOGOUS    PROPERTIES    OP 


From  the  extremity  of  the  latus-rectum  LST  draw  LK  at  right 
<  s  to  the  directrix ;  and  from  p  draw  pm  parallel,  and  pn  perp'en- 


M 

Ts 

L 

p 

K 

1  n 

ft 

aI 

s 

in 

P 

T\ 

1 

\^v 

dicular,  to  the  axis.     From  the  nature  of  the  construction  we  have, 
SP  :  PM  (or  NE)  : :  SL  :  LK  (or  SE), 
or  SP  :  SL  : :  NE  :  SE  ; 
.-.  SP— SL  :  SL  : :  NE— SE  (SN)  :  SE, 
i.  e.  SP— SL  :  SN  : :  SL  :  SE  ; 
for  the  same  reason,  ST— S/>  (SL— S^) :  Sn  : :  SL  :  SE  ; 

.-.  SP— SL  :  SL— $p  : :  SN  :  Sn  : :  (by  sim.  As)  SP  :  Sp. 

Hence  SPxS/?— SLxSp=SL  xSP— SPxSp, 

or  2SPxSp=SL  x(SP+Sp)=SLxPp ; 
.-.  4SPxSp=2SLxPp=latus-rectumxPp. 


Cor.  1.   Hence  p+^| 


(151.)   Cor.  1.   Hence  fp+o-=dT-*    For  since  SLx(sp+sP)= 

ocm    a  u         SP+S^/       1    ,    1  \       2 

2SPxSP;  we  have  g^     +_)-_. 


THE   THREE   CONIC   SECTIONS.  .  Ill 

Cor.  2.  Since  SP— SL  :  SL— Sp  : :  SP  :  Sp,  SP,  SL  and  Sp  are 
in  harmonical  proportion.  Or,  half  the  latus-rectum  is  an  harmo- 
nical  mean  between  the  segments,  into  which  the  focus  of  a  conic 
section  divides  any  line  which  passes  through  it. 

XVII. 

On  the  analogous  Properties  of  the  Normal,  Latus-rectum,  Radius 
of  Curvature,  $*c.  $*.  in  all  the  Conic  Sections. 

If  S  be  the  focus,  A  the  vertex,  and  P  any  point  in  the  Parabola, 
then  (Arts.  125,  126.)  4SP= chord  of  curvature  passing  through 

3  3 

4SP^  2SP2 

the   focus :  —^-rr  =  diameter,  and  -t-&~t  «=  radius  of   curvature 

V(SA)  V(bA) 

to  the  same  point.  In  the  Ellipse  and  Hyperbola,  if  C  be  center,  S 
the  focus,  AC  the  semi-axis  major,  CD  the  semi-conjugate  to  the 
semi-diameter  PC,  and  PF  a  perpendicular  let  fall  from  the  point  P 

2CD2 

to  the  conjugate  diameter,  then  (Arts.  128,  129.)  -p~-  =  chord  of 

2CD2 
curvature  passing  through  the  center  from  the  point  P ;  — r^r  =  chord 

through  focus  ;  -p^r  =  diameter,  and  p^~  =  radius  of  curvature 
to  the  same  point. 

Property  1. 

In  every  conic  section,  the  cube  of  the  normal  divided  by  the  ra- 
dius of  curvature  is  equal  to  the  square  of  half  the  latus-rectum. 

(152.  In  the  Parabola,  (see  Fig.  in  page  30.)  since  normal  PO  : 
SY::TP:TY,    and    TP  =  2TY,    .-.PO  =  2SY;   hence  cube  of 

normal  =  8SY3  =  (by  Art.  32.)  8SP*  x  SA^  ;   radius  of  curvature 

2SP^  cube   of  normal         8SP*xSA*xSA* 


f,A  rad.  of  curvature  2SP£ 

square  of  2S A  =  square  of  half  the  latus-rectum. 


=  4SA2 


112  *     ANALOGOUS    PROPERTIES    OP 

(153.)  In  the  Ellipse  and  Hyperbola  (see  Figures  and  Properties 

RC2 
in  pp.  49,  67,  68.)  POxPF=BC3;    /NI»©-5=r,   and    cube    of 

,      BC8        *S       i  CD2    .  cube  of  normal 

normal  =  5==- ;  radius  of  curvature  =.™-;  hence  — = — ? — 

PF 3  PF  rad.  of  curvature 

=  CD^PF^=CD2^PF2==(by    Pr°pertieS    m    PP-     5L     69') 

BC«  BC*  ,BC2  _,    ..  .     ,  , 

square  of  -r— r=square  of  half  the  latus-rectum. 


AC3xBC2      AC2        u AC 

(154.)  Cor.  Since  half  the  latus-rectum  is  a  constant  quantity,  the 
radius  of  curvature  varies  as  the  cube  of  the  normal ;  the  curvature 
therefore  varies  inversely  as  the  cube  of  the  normal  in  all  the  Conic 
Sections  ;  which  accords  with  what  has  already  been  demonstrated 
in  Sections  XIII  and  XIV. 

Property  2. 

In  any  conic  section,  if  a  perpendicular  (OX)  be  let  fall  upon  the 
line  SP  from  the  point  O,  where  the  normal  intersects  the  axis,  then 
the  part  PX  cut  off  by  this  perpendicular  is  equal  to  half  the  latus- 
rectum. 

(155.)  In  the  Parabola. 
Draw  the  ordinate  PN ;  then, 
since  (by  Art.  30.)  SP  =  SO, 
the  angle  SOP  =  SPO,  and 
PO  is  common  to  the  two 
right-angled  triangles  PXO, 
PON  ;  these  two  triangles  are 

therefore  equal  and  similar ;  T  A    S        N  o 

hence  PX  =  NO  =  (Art.  21.)  half  the  latus-rectum. 

(156.)  In  the  Ellipse  and  Hyperbola.     (Fig.  in  p.  113.) 
Draw  the  conjugate  diameter  DCK,  then  the  right-angled  trian- 
gles PEF,  PXO  are  similar;  .-.PE(AC)  :  PF  ::  PO  :  PX;  hence 

"PC2 
AC  x  PX  -  PO  x  PF  -  BC2 ;  .-.  PX  =   ~  =  half  the  latus-rectum. 


THE    THREE    CONIC    SECTIONS. 


113 


(157.)  Cor.  By  means  of-  this  Property,  if  SP  be  given  in  length 
and  position,  and  the  latus-rectnm  and  position  of  the  tangent  be  also 
given,  we  can  determine  geometrically  the  position  of  the  axis  ;  for 
we  have  only  to  make  PX  equal  to  half  the  latus-rectum,  and  draw 
XO  at  right  angles  to  SP,  and  PO  at  right  angles  to  the  tangent  at 
P,  then  O  (the  intersection  of  XO,  PO)  is  a  point  in  the  axis,  which 
being  joined  to  S,  gives  SO  the  position  of  the  axis. 


Property  3. 

In  any  conic  section,  take  the  arc  PQ,,  and  from  the  point  Q,  draw 
GIT  perpendicular  and  Q,R  parallel  to  SP  ;  then  (assuming  the  rela- 
tion of  Q,T  and  Q,R  just  at  the  point  of  contact)  the  latus-rectum  is 

Q,Ta 

equal  to -^ 


(158.)  In  the  Parabola.'  Draw  the 
perpendicular  SY  upon  the  tangent 
PY;  then,  since  the  arc  Q,P  coincides 
with  the  tangent  at  P,  the  triangle 
Q,PT  continually  approaches  towards 
similarity  with  the  triangle  SPY  as  Q, 
moves  up  towards  P  ;  and  at  the  point 
of  contact  QP  :  QT  :  :  SP  :  SY ; 
.-.  QP2  :  QT2  : :  SP2  :  SY2,  and   (dividing  the  first  two  terms  by 

C.  S.  15 


114  ANALOGOUS    PROPERTIES    OF 

QR)^  :~::  SP2  :  SY2  ::  (by   Art.   32.)  SP2  :  SPxSA  :  : 

Q,P2 

SP  :  SA.   Now  -^-=-(=chord  of  curvature  passing  through  the  fo- 

cus)  =  4SP;    hence  we    have  4SP    :    ^=-::SP.-.SA,    :  ^— = 

4SPxSA     ACj.      .  ; 

— —  - — =  4S  A  =  latus-rectum. 

(159.)  In  the  Ellipse  and  Hyperbola.     Draw  the  conjugate  diam- 
eter DCK,  and  the  perpendiculars  PF  and  SY  upon  it  and  the  tan- 


gent ;  then  the  triangles  QPT,  PEF  are  similar,  .-.  QP2  :  Q/T2  : : 

PE2  (AC2)  :  PF2,  and  ^~  (-jfr^**^  l~i;  AC2   :   PF5 
Q,K  V  AG  /       U,ri 

QT2    2CD2xPF2    2AC2xBC2    2BC2 


QR  AC3  AC  ^  AC 


the  latus-rectum. 


The  demonstration  of  this  property  of  the  Conic  Sections  forms 
the  substance  of  the  first  three  Propositions  of  the  third  Section  (B. 
1 .)  of  Sir  Isaac  Newton's  Principia. 

Property  4. 

(160.)  In  every  conic  section,,  the  chord  of  curvature  passing 
through  the  focus  is  to  the  latus-rectum  in  the  duplicate  ratio  of 

i 


THE    THREE    CONIC    SECTIONS.  115 

SP  :  SY  ;  and  the  diameter  of  curvature  is  to  the  same  in  the  trip- 
licate ratio  of  SP  :  S  Y. 

QP2 

For  the  chord  of  curvature  passing  through  the  focus  =  ;=^5-;  and 

Q,T2 

by  Property  3,  the  latus-rectum  =  ?-=- ;  hence  the 

,     ,    *  "  &P2    T&2 

chord  of  curvature  :  latus-rectum  : :  — — - :  — — —  : :  GlP2  :  Q.T2  : :  SP2  :  SY2  : 

(dK.      CIR 

but  diameter  :  chord  of  curvature  (see  Fig.  in  page  95.)  : :  SP  :  SY ; 

.'.  diameter  of  curvature  :  latus-rectum  : :  SP*  :  SY3  . 


Property  5. 

Let  L  =  latus-rectum  of  any  conic  section  ;  then,  in  the  Parabola, 
LxSP-=4SY2;  in  the  Ellipse,  LxSP  is  less  than  4SY2 ;  and  in 
the  Hyperbola,  L  xSP  is  greater  than  4SY2. 

(161.)  In  the  Parabola.  (32)  SAxSP  =  SY2,  .-.4SAxSP=4SY2, 
or  LxSP=4SY2,  for  L=4SA. 

RC2  v  SP 
(162.)  In  the  Ellipse.     By  Art.  66.  SY2  =        *       ,    ...  4SY2 

HP 

4BC2  x  SP      /  ,      2BC2 


HP 


=  (  for  -xc  ~h'    and    ' ' ' 4BC2  =  2AC  X  L  ) 

^^— P  5  hence  L  x  SP  :  4SY2  : :  HP  :  2AC  : :  2AC— SP*  : 

2 AC,  and  as  2 AC — SP  is  less  than  2 AC,  L  x  SP  must  be  less  than 
4SY2. 

(163.)  In  the  Hyperbola,  by  a  similar  process  we  have  LxSP  : 
4SY2  : :  HP  :  2AC  : :  2AC+SPt  :  2AC,  and  as  2AC+SP  is  greater 
than  2AC,  LxSP  must  be  greater  than  4SY2. 


*  For  SP+HP=2AC,  .-.  HP=2AC— SP. 
t  For  HP— SP=2AC,  ..  HP-2AC+ SP. 


116  ANALOGOUS   PROPERTIES,   &C. 

(164.)  Before  we  conclude  this  Section,  it  will  be  proper  to  show 
the  method  of  expressing  the  relation  between  SP  and  SY,  in  the 
form  of  an  algebraic  equation.  In  the  Parabola,  therefore,  let  SA— 
a,'  SP=x,  SY=y;  then  since  SY2=SAxSP,  we  have  y2=ax,  or 
y=V(ax,)  for  the  equation  to  the  curve,  in  terms  of  the  distance 
from  the  focus,  and  the  perpendicular  from  the  focus  upon  the  tan- 
gent.    In  the  Ellipse  and  Hyperbola,  let   AC=a,   BC=^  SP=:r, 

SY  =  y;    then  HP  =  2AC±SP=2a±;r,  .-.since  SY2  =  — -- — , 

b2  x  b2x2  bx 

we  have   y  -  ^—  =  g-^  and  y-  v(2g^  where  the 

negative  or  positive  sign  must  be  used  according  as  the  section  is  an 
Ellipse  or  an  Hyperbola.* 

(165.)  To  investigate  the  relation  between  CP  and  Cy  (see  Fig- 
ures in  pages  49,  68,)  let  CP=x,  Cy  or  PF=y ;  then  in  the  Ellipse, 
since  AC2  +  BC2  =  CD2  +  PC2,  we  have  a2  -{-  b2  =  CD2  +  x2, 
...  CD2  -  a2  +  b2—x2    or    CD  =  V  (a2  +  b2— x2 .)       Again,     since 

ACxBC=CDxPF,   we   have   ab  =  CT)xy,  .-.  CD=  — ;  hence 

—  =  V(a2-f62 — x*,)  or  y  =  — — — — — is  the  equation  to  the 

y         K  '  V(a2  +  Z>2— x2)  * 

curve  in  terms  of  the  distance  from  the  center,  and  'perpendicular 
from  the  center  upon  the  tangent. 

In  the  Hyperbola,  PC2  ^CD2  =  AC2  ^CB2,  or  #2^CD2  =  a2^b2 ; 

,-.  CD2  =  *2-a2+&2,  and  y-  "V^-t 

V  (#2 — a2-fo2) 


*  These  expressions  are  the  equations  of  the  several  conic  sec- 
tions, considered  as  spirals,  described  by  the  revolution  of  the  radius 
vector  SP,  about  the  focus. 

t  In  these  equations,  the  curves  are  considered  as  described  by  a 
radius  vector  CP,  revolving  about  the  center.  This  mode  of  con- 
sideration is,  of  course,  inapplicable  to  the  Parabola. 


METHOD    OF   FINDING,  &C. 


117 


CHAPTER  VII. 

ON  THE  METHOD  OP  FINDING  THE  DIMENSIONS  OF  CONIC 
SECTIONS  WHOSE  LATERA-RECTA  ARE  GIVEN,  AND  OF 
DESCRIBING  SUCH  AS  SHALL  PASS  THROUGH  CERTAIN 
GIVEN  POINTS. 


XVIII. 

On  the  method  of  finding  the  dimensions  of  Conic  Sections,  whose 
later a-recta  are  given. 

(166.)  Let  S  be  the  focus  of  any  conic  section,  P  some  point 
in  the  curve  at  a  given  distance  from  S  ;  join  SP,  and  let  it  meet  the 
tangent  PT  in  the  given  angle 
SPT  ;  let  the  latus-rectum  = 
L,  and  take  PX=£L  ;  from  X 
draw  XO  at  right  angles  to  SP, 
and  from  P  draw  PO  at  right 
angles  to  PT,  then  by  Art.  157, 
O  will  be  a  point  in  the  axis 
join  SO,  and  it  will  give  the  po 
sition  of  the  axis. 

(167.)  We  are  thus  furnished  with  the  means  of  determining  geo- 
metrically the  position  of  the  axis  of  any  conic  section  whose  latus- 
rectum  is  given,  and  whose  tangent  at  a  given  point  meets  a  line 
drawn  from  the  focus  to  that  point,  in  a  given  angle.  The  position 
of  the  axis  being  found,  its  dimensions  may  be  ascertained  from  the 
properties  of  each  particular  curve.  In  the  Parabola,  the  latus-rec- 
tum  is  equal  to  four  times  the  distance  of  the  focus  from  the  vertex  ; 
if  therefore  in  OS  produced,  we  take  SA  equal  to  £L,  A  will  be  the 
vertex  of  the  Parabola.  In  the  Ellipse  and  Hyperbola,  it  will  be 
necessary  to  find  the  center,  as  also  the  major  and  minor  axis ;  which 
is  done  in  the  following  manner. 


118 


METHOD  OF  FINDING  THE 


(168.)  In  the  Ellipse,  the  lines  drawn  from  the  foci  to  any  point 
in  the  curve  make  equal  angles  with  the  tangent  at  that  point ;  if 
therefore  the  angle  HPZ  be  made  equal  to  the  angle  SPY,  and  SO 
be  produced  to  meet  PH  in  the  point  H,  that  point  will  be  the  other 
focus ;  and  this  determines  the  length  (SP+PH)  of  the  major  axis. 
Now  by  Art.  45,  the  conjugate  diameter  DCK  cuts  off  from  SP  a 
part  equal  to  the  semi-axis  major ;  hence  if  PE  be  taken  equal  to 
J(SP-l-PH),  and  through  E  we  draw  DC  parallel  to  the  tangent  at 
P,  C  will  be  the  center  of  the  ellipse.  It  only  remains  therefore  to 
produce  SH  both  ways,  and  make  CA,  CM  each  equal  to  PE,  and 
we  have  AM  the  major  axis  of  the  curve.  But  (39)  the  latus-rec- 
tum  is  a  third  proportional  to  the  major  and  minor-axis  ;  the  minor 


axis  is  therefore  a  mean  pro- 
portional between  the  major 
axis  and  the  latus-rectum ; 
from  C  then  draw  BC  at  right 
angles  to  AM,  make  BC  a  A 
mean  proportional  between 
AC  and  ^L,  and  B  will  be  the 
extremity  of  the  minor  axis  ; 
thus  the  dimensions  of  the  el- 
lipse are  determined. 


BY 


(169.)  In  the  Hyperbola,  the  tangent  bisects  the  angle  SPH;  in 
this  case,  therefore,  the  angle  HPY  must  be  made  equal  to  the  an- 
gle SPY  on  the  opposite  side  of  the*  tangent ;  then  if  OS  is  produced 
till  it  meets  PH  in  Ihe  point  H,  that  point  will  be  the  other  focus. 
Produce  SP  to  E,  and  take  PE  equal  to  J(HP— SP) ;  through  E 


DIMENSIONS    OF    CONIC    SECTIONS. 


119 


draw  EC  parallel  to  the  tangent  at  P,  and  C  will  be  the  center. 
Take  OA,  CM,  each  equal  to  PE,  then  AM  will  be  the  major  axis. 
The  minor  axis  is  determined  precisely  in  the  same  manner  as  in 
the  Ellipse. 

(170.)  We  have  thus  shown  the  method  of  solving  this  Problem, 
when  the  nature  of  the  curve  is  given.  Suppose  now  that  the  latus- 
rectum,  the  distance  SP,  and  the  position  of  the  tangent  be  given  as 
before,  and  it  is  required  to  find  not  only  the  dimensions,  but  the  na- 
ture of  the  conic  section.  In  this  case  we  have  recourse  to  Arts. 
161,  162,  163  ;  from  which,  when  the  latus-rectum  and  the  relation 
between  SP  and  SY  are  given,  we  can  determine  the  particular  na- 
ture of  the  curve.  For  it  is  there  proved,  that  if  LxSP  be  equal  to 
4SY2,  the  curve  is  a  Parabola ;  if  LxSP  be  less  than  4SY2,  it  is 


H      M 


AS       N 


H  M 


an  Ellipse ;  and  if  LxSP  be  greater  than  4SY2,  it  is  an  Hyperbola. 
In  order  to  affect  this  general  solution  of  the  Problem,  let  the  sine  of 
the  given  <  SPY=s,  radius=l,  then  (by  Trigonometry)  SP  :  SY  : : 
1:5;  .-.  SY=s  .  SP,  and  SY2=s2 .  SP2 ;  consequently  4SY2=4s2 . 
SP2.  Having  therefore  found  the  position  of  the  axis,  as  in  the 
former  case ;  then,  to  know  whether  the  conic  section,  whose  di- 
mensions are  required,  be  a  Parabola,  Ellipse,  or  Hyperbola,  we 
must  compare  LxSP  with  4s2 .  SP2.  If  LxSP  be  equal  to  4s2 . 
SP2,  i.  e.  if  L  be  equal  to  4s2 .  SP,  then  the  curve  is  a  Parabola; 
take  therefore  SA=|L,  and  A  is  the  vertex.  If  L  be  less  than  4sf  . 
SP,  the  curve  is  an  Ellipse  ;  in  which  case,  make  the  <  HPZ  (on 
the  same  side  of  the  tangent  with  SP)  equal  to  SPY,  and  proceed 
as  in  Art.  168.  If  L  be  greater  than  4s2 .  SP,  the  curve  is  an  Hy- 
perbola ;  make  therefore  HPY  (on  the  other  side  of  the  tangent) 
equal  to  SPY,  and  proceed  as  in  Art.  169. 


120  METHOD  OF  FINDING  THE 

(171.)  By  Art.  160,  the  chord  of  curvature  passing  through  the 
focus  :  the  latus-rectum  : :  SP2  :  SY2  : :  1  :  s2 ;  .-.  the  latus-rec- 
tum =s2x  chord  of  curvature  ;  if  therefore  the  chord  of  curvature 
and  the  relation  of  SP  to  SY  be  given,  the  latus-rectum  will  also  be 
given.  We  are  thus  enabled  to  give  the  trigonometrical  solution  of 
the  following 

PROBLEM. 

(172.)  Given  the  chord  of  curvature  passing  from  any  point 
through  the  focus  of  a  conic  section,  the  distance  of  that  point  from 
the  focus,  and  the  position  of  the  tangent ;  it  is  required  to  find  the 
nature  and  dimensions  of  the  conic  section. 

e 

Let  the  chord  of  curvature  to  the  point  P=40,  SP=12,  the  angle 
SPY=30° ;  then  since  the  sine  of  30°=half  radius,  s=£ ;  .-.  L«= 
(s2X  chord  of  curvature  =)£  x  40=10  ;  also  4s2  xSP  =  4  x^SP  = 
SP=12  ;  hence  L  is  less  than  4s2xSP,  and  consequently  the  conic 
section  is  an  Ellipse. 

Z 


Since  the  <  SPY  -=  30°,  the  <XPO  =  60;  .-.  <XOP  =  30°, 
and  PX  =  |PO,  or  PO  =  2PX  =  (Art.  156.)L  =  10.  Hence,  in  the 
triangle  SPO,  we  have  SP  =  12,  PO  =  10,  <  SPO  =  60°  from 
which  we  can  determine  the  <  PSO  ;  for  POS  +  PSO  =  120°,* 
.-.  i(POS  +  PSO)=60°.     Now  SP  f  PO  (22)  :  SP— PO  (2)  : :  tan. 

KPOS+PSOX60O)  :  tan.  ^08^80)  =  ^^=^ 

.-.log.  tan.  }  .  (POS— PSO)=log.  tan.  60°— log.  ll=log.tan.  8°  57' ') 
Ijence  <  PSO  =  (|(POS +PSO)-i(POS— PSO)  =  )60°— 8°  57'= 
51°  3'. 

Since  <HPZ=  <  SPY,  the  <OPH=  <  SPO,  .-.  <  SPH  = 
120° ;  in  the  triangle  SPH  we  have  therefore  SP=12,  <  PSH= 

*  See  Day's  Trigonometry,  Art.  153. 


DIMENSIONS    OF    CONIC    SECTIONS. 


121 


5°  57' 
:  PH  = 


;  but  as  sin.  PHS 
12  x  sin.  51°  3' 
sin.  8°~5f       ' 


8°  57'= log. 


51°  3',  <SPH  =  120°,  and  .-.  <PHS  = 

(8°  57')*  :  sin.  PSH  (51°  3')  : :  SP  (12) 

hence  log.  PH=log.  12-flog.  sin.  51°  3' — log.  sin 
59.987;  .-.  PH=  59.987,  and  SB+PH  =  12+59.987  =  71.987- 
major  axis  of  the  ellipse,  and  the  minor  axis=  (mean  proportional  be- 
tween the  major  axis  and  latus-rectum=)  V  (10x71.987)  =  26.83; 
from  which  the  Ellipse  may  be  constructed  as  in  Art.  168. 

XIX. 

On  the  method  of  describing    Conic  Sections  which  shall  pass 
through  three  given  points. 

(173.)  Let  SO,  SP,  SGI,  be  three  lines  given  in  length  and  posi- 
tion ;  join  PO,  Q,P  ;  produce  PO  to  p,  making  Op  :  Vp  : :  SO  :  SP ; 
and  produce  it  both  ways  to  m  and  D.  Draw  SE,  On,  PM,  Qm,  at 
right  angles  to  mED ;  then  the  conic  section  whose  focus  is  S,  di- 
rectrix MED,  and  determining  ratio  SO  : :  On,  will  pass  through 
the  points  O,  P,  Q. 


(174.)  By  sim.  As  Onp,  PM/?,  Op 
the  construction  Op  :  Vp  : :  SO  :  SP,  . 


;  Pp  : :  On  :  PM ;  but  by 
SO  :  SP  : :  On  •  PM,  and 


as. 


See  Day's  Trigonometry,  Art.  150. 
16 


122  ON   CONIC*  SECTIONS 

SO  :  On  : :  SP  :  PM.  Again,  by  sim.  A  s  FMq,  QLmq,  Vq:Q,q:: 
PM  :  am ;  but  Vq  :  Q?  : :  SP  :  SQ,-.SP  :  Sa  : :  PM  :  Qra,  or  SP  : 
PM  : :  Sa  :  dm ;  hence  SO  ;  On  : :  SP  :  PM  :  :  SQ,  :  am,  i.  e.  the 
lines  SO,  SP,  SO,  diverging  from  S  are  in  a  given  ratio  to  the  lines 
On,  PM,  Q,ra  drawn  at  right  angles  to  the  line  MED.  By  Art.  138, 
therefore  the  curve  OPQ,  is  a  conic  section  whose  focus  is  S  and  di- 
rectrix MED ;  and  it  will  be  a  parabola,  ellipse,  or  hyperbola,  ac- 
cording as  the  antecedent  of  that  ratio  is  equal  to,  less  or  greater 
than,  the  consequent,  or  according  as  SO  is  equal  to,  less  or  greater 
than  Ora. 

(175.)  In  order  to  find  the  dimensions  of  the  conic  section  ;  divide 
SE  at  the  point  A,  so  that  SA  :  AE  : :  SO  :  On,  and  A  will  be  the 
vertex.  If  SO  — Ow,  then  SA=AE,  and  the  curve  is  a  Parabola 
whose  axis  is  EAS,  vertex  A,  and  latus-rectum  4SA.  If  SO  be 
less  than  On,  take  AC  :  SC  : :  On :  SO,  then  (by  Sect.  XVI.)  C 
will  be  the  center  and  AC  the  semi-axis  major  of  the  Ellipse ;  the 
semi-axis  minor  (BC)=V(AC2 — SC2.)  If  SO  be  greater  than 
On,  take  C  on  the  other  side  of  A,  so  that  AC  :  SC  : :  On  :  30, 
then  C  will  be  the  center,  and  AC  the  semi-axis  minor  (BC)  = 
V(SC2— AC2.) 

This  method  of  construction  leads  to  the  trigonometrical  solution 
of  the  following 

PROBLEM. 

(176.)  Three  straight  lines  issuing  from  a  point,  being  given  in 
length  and  position  ;  it  is  required  to  find  the  nature  and  dimensions 
of  the  conic  section  which  shall  pass  through  the  extremities  of  those 
three  straight  lines,  and  have  its  focus  in  the  point  of  their  inter- 
section. 

Let  SO  =4,  SP-7,  SGI  =  10,  <  OSP=60°,  <  PSQ  =  20°; 
then  SOP+SPO=120°,  and  SP+SO  (11)*  .  SP— SO   (3)  : :  tan. 


*  See  Day's  Trigonometry,  Art.  153. 


PASSING   THROUGH    THREE    POINTS. 

3  x  tan.  60° 


11 


123 


log. 


*  (SOP  +  SPO)  (60°)  :  tan.  i  (SOP— SPO) 
tan.  )  .  (SOP— SPO)=log.  3+log.  tan.  60°— log.  ll=log.  tan.  25° 

Q 


17';  hence  <  SOP=60°  +  25°  17'  =  85°  17',  and  <  SPO =  60°— 
25°  17'=34°  43'. 
Now,  sin.  SPO  (34°  43')*  :  sin.  OSP  (60°) : :  SO  (4)  ;  PO  = 


4  x  sin.  60° 


.-.log.  PO=log.  4+ log.  sin  60°— log.  sin.  34c43'= 


sin.  34°  43'' 

log.  6.0825,  .*.  PO=6.0825.  By  a  similar  process  it  appears  that  the 
angle  SPQ=125°  1',  <  SQP=34°  59',  and  PQ=4.1758.  Thus  all 
the  sides  and  angles  of  the  triangles  SOP,  SPO,  are  determined, 
and  we  now  proceed  to  find  the  values  of  Pp  and  Qq. 

By  the  construction  Op  :  Vp  : :  SO  :  SP  : :  4  :  7 ; 
.-.  Op  :  Vp—Op  (OP)  : :  4  :  3. 
Hence  0^=f  OP=f  x  6.0825=8.11 ; 

...  Pp=Op+OP=6.0825+8.11=14.1925. 
Again,  Vq  :  Qq  : :  SP  :  SQ  : :  7  :  10  ; 

...  p^  :  Qq—Vq  (PQ)  : :  7  :  3. 


See  Day's  Trigonometry,  Art.  150. 


124  ON   CONIC    SECTIONS 

Hence  Vq  =  J  x  Pa  =  J  X  4.1758  -  9.7435  ; 

...  Qq =Vq  +PQ  =  9.7435  +4.1758  - 13.9193. 

Now  the  <SP?=180°— SPa=180°— 125°  l'  =  54°  59';  and 
the  <  pVq  =  S¥q— SPO  =  54°  59'— 34°  43' =  20°  16';  hence,  in 
the  triangle  qVp,  we  have  Fq  =  9.7435,  Vp  =  14.1925,  and  the  in- 
cluded angle  pVq  =  20°  16'-;  from  which  the  angle  qpF  is  found  to 
be  equal  to  33°  45'.  In  the  right-angled  triangle  Opn,  we  have 
therefore  Op  =  8.11,  and  the  <  Op?i  =  33°  45',  which  gives*  Ora  = 
4.5056. 

Again,  in  the  triangle  PpM,  we  have  Vp  =  14.1925,  and  the  angle 
P^M  =  33°  45',  from  which  PM  is  found  to  be  equal  to  7.8849. 
Finally,  by  similar  triangles,  P^M,  Qqm,  we  have  P^  (9.7435)  : 

Q?  (13.9193)  : :  PM  (7.8849)  :  am=^-|^==  11.264. 

On  reviewing  the  steps  of  this  operation,  we  have, 
SO  :  On  ::    4  :    4.5056  : :  1  :  1.1264, 
SP  :  PM::    7  :    7.8849  ::  1  :  1.1264, 
SQ  :  am::  10   :  11.264    ::  1  :  1.1264.t 
The  given  ratio  therefore  of  SO  :  On  is  1  :  1.1262;  and  as,  in 
this  case,  SO  is  less  than  On,  the  curve  is  an  Ellipse. 


Having  thus  ascertained  the  nature  of  the  curve,  it  now  only  re- 
mains to  find  its  dimensions.  For  this  purpose  we  must  first  find 
the  length  of  SE,  which  (if  ON  be  let  fall  perpendicular  to  it)  is 
equal  to  SN  +NE,  i.  e.  to  SN  +  On,  for  On  is  equal  to  NE,  being 
the  opposite  side  of  a  parallelogram. 

Now  <  NOS  =  180°  —  <  SOP  —  <  NOp  (Opn)  - 180°— 85°  17' 
—  33°  45' =  60°  58'. 


*  See  Day's  Trigonometry,  Art.  134. 

t  It  is  not  necessary  to  find  all  three  of  these  ratios,  since  they  are 
equal  to  one  another.  It  would  have  been  sufficient  to  have  calcu- 
lated the  length  of  On,  merely  ;  for  the  ratio  SO  :  On  determines  the 
nature  of  the  curve. 


PASSING   THROUGH   THREE    POINTS.  125 

In  the  triangle  OSN  we  have  therefore  OS=4,  and  the  angle 
SON=60°  58',  from  which  we  get  SN=3.4973  ;  and  consequently 
SE(=SN+Orc)=3.4973-f-4.5056=8.0029. 

Having  found  the  value  of  SE,  we  must  divide  it  in  the  ratio  of 
SO  :  On ;  i.  e.  in  the  given  ratio  of  1  ;  1.1264. 

Thus  SA  :  AE  : :  OS  :  On  : :  1  :  1.1264 ; 

.-.  S A  :  SA  -f  AE(SE)   : :  1  :  2.1264. 

Hence  SA=^— m==  '         =3.7635  ;  therefore,  make  the  angle 

OSA=90°— 60°  58'=29°  2',  and  take  SA=3.7635,  then  A  will  be 
the  extremity  of  the  major  axis. 

To  find  the  major  axis  itself,  take 

AC  :  SO  : :  On  :  OS  : :  1.1264  :  1, 
or  AC  :  AC— SC(AS)    : :  1.1264  :  .1264  ; 
.£     1.1264     ACJ 
•'•AC~T1264XAS' 

=  ^^7  X  3.7635=33.4951. 
.1264 

Hence  SC=AC—AS=33.4951— 3.7635-29.7316. 

and  BC=V(AC2— SC2)=15.417. 

Finally  the  dimensions  of  the  Ellipse  are  as  follow  : 

Major  Axis=2AC=66.9902. 

Minor  Axis=2BC=30.834. 

2BC2 
Latus-rectum= ——^=14.185. 

By  means  of  this  Problem  the  dimensions  of  the  orbit  of  a  Planet 
or  Comet  may  be  found  from  three  observations  made  as  to  its  dis- 
tance and  angular  position,  at  three  different  periods  in  the  course 
of  one  revolution  round  the  Sun. 


126 


QUADRATURE   OF   THE   CONIC   SECTIONS. 


CHAPTER  VIII. 
ON  THE  QUADRATURE  OF  THE  CONIC  SECTIONS. 


XX. 


On  the  relation  which  obtains  between  the  areas  of  Conic  Sections 
of  the  same  kind,  having  the  same  vertex  and  axis  ;  and  on  the 
Quadrature  of  the  Parabola,  Ellipse,  and  Hyperbola. 

(177.)  Let  AQq,  APp,  be  any  two  curves  having  the  same  ver- 
tex A,  and  let  them  be  referred  to  the  same  axis  AM  by  ordinates 


.B 

Q 

\m 

rB 

>b        , 

V 

°\ 

•b 

M 


N 


If 


GIN,  PN,  which  are  to  each  other  in  a  given  ratio  ;  then  the  areas 
AGIN,  APN,  generated  by  those  ordinates,  will  be  to  each  other  in 
the  same  given  ratio.  For  take  the  ordinate  qpn  indefinitely  near  to 
Q,PN,  and  draw  dm,  Vo  parallel  to  the  axis,  then  the  flnxional  or 
incremental  areas  QNn$  PNrcjo  will  approach  to  equality  with  the 
parallelograms  QJSnm,  PNno,  as  the  ordinate  qpn  approaches  to 
Q,PN ;  but  (by  Euc.  1.  6.)  these  parallelograms  are  to  each  other 
in  the  ratio  of  GIN  :  PN ;  the  nascent  increments  therefore  of  the 
areas  AGIN,  APN  are  to  each  other  in  the  ratio  of  GIN  :  PN  ;  and 
as  these  areas  begin  together  from  A,  the  areas  themselves  must  also 


QUADRATURE    OF   THE    CONIC   SECTION.  127 

be  to  each  other  in  that  ratio,  i.  e.  area  AGIN  :  area  APN : :  GIN  : 
PN. 

(178.)  Suppose  now  the  curves  AQq,  APp  to  be  two  Conic  Sec- 
tions of  the  same  kind  whose  latera-recta  are  respectively  L  and  I ; 
for  instance,  let  them  be  two  Parabolas  ;  then  by  the  property  of  the 
parabola  LxAN  =  QN2,  and  ZxAN  =  PN2,  hence  AQN  :  APN 
(::QN  ;  PN)  : :  V (LxAN)  :  V(ZxAN)::VL  :  W.  If  they  be 
Ellipses  or  Hyperbolas  which  have  the  same  major  axis  AM,  and 
whose  minor  axes  are  respectively  BC  and  bC,  then 

'       ANxNM  :        PN2     : :  AC2  :  6C2, 
and  GIN2  :    ANxNM : :  BC2  :  AC2  5 

or>r<2      Q.hC*2 

.-.QN2  :        PN2     ::BC2  :&C2" 


AC    *   AC 
:  :L  :  I 

Hence  in  this  case  also  AGIN  :  APN( : :  aN  :  PN) : :  VL  :  Vl ; 
i.  e.  in  Conic  Sections  of  the  same  kind,  having  the  same  vertex  and 
axis,  the  areas  AGIN,  APN  are  to  each  other  in  the  given  subdupli- 
cate  ratio  of  their  latera-recta. 

(179.)  Take  any  point  S  in  the  axis,  and  join  SGI,  SP ;  then  we 
have 

area  AQN    :  area  APN : :  QN  :  PN, 

andASaN    :  A  SPN     ::QN:PN; 


.•.AQN— SQN  :     APN— SPN  ::GtN  :  PN, 
or  area  AGIS  :  area  APS  : :  GIN  :  PN 

::VL;  Vl  in    all    the   Conic 
Sections. 

(180.)  But  Ellipses  and  Hyperbolas  having  the  same  vertex  and 
axis,  will  also  have  the  same  center.*  Let  C  be  that  center,  and  in 
each  case  join  GIC,  PC ;  then 


*  Although  they  have  the  same  center,  it  should  be  recollected 
that  (since  the  minor  axes  are  not  equal)  they  cannot  have  the  same 
focus. 


128 


QUADRATURE    OF    THE    PARABOLA. 


In  the  Ellipse,  AQN  :     APN    : :  QN 

and  A  QCN  :  A  PON  : :  QN 

.-.  AQN+QCN  :  APN+PCN  : :  QN 

or  sector  ACQ, :  sector  ACP  : :  QN 

::VL 


PN, 
PN; 
PN, 
PN, 

Vl. 


In  the  Hyperbola,  AQN  :      APN  : :  QN  :  PN, 

and  A  QCN  :  A  PCN  : :  QN  :  PN  ; 

.-.  QCN— AQN  :  PCN— APN  : :  QN  :  PN, 

or  sector  ACQ  :  sector  ACP  : :  Q^N  :  PN, 

::  VL  :  Vl. 

On  the  Quadrature  of  the  Parabola. 


(181.)  Let  AP  be  a  Parabola  whose  latus-rectum  (BC)=a,  ab- 
scissa (AN)  =#,  ordinate  (PN)  =  y  ;  then,  by  the  Property  of  the 

Parabola,  yz=ax,  .:y=a2x2j  and    yX=a2x2'x   now  the  fluent  of 

%xYa2x2=\xy\  hence  the  area  APN=fANxPN=f 


yx 


=  %a2x2= 


the  circumscribing  parallelogram. 


QUADRATURE  OF  THE  PARABOLA.  129 

(182.)  Draw  a  tangent  to  the  point  P,  and  produce  NA  to  meet 
it  in  T  ;  then  since  AN  =  |NT,  the  A  PNT  =  (£TNxPN=  )AN  x 
PN;  hence  the  area  ANP=§ANxPN=f  A  PNT.  Now  sup- 
pose, in  the  Figure,  at  page  28,  that  a  tangent  be  drawn  to  the  point 
G,  and  that  the  line  MS  drawn  parallel  to  the  axis  meets  it  in  S, 
then  the  area  AZG  =  |A  TZG  ;  but  A  TZG  :  A  SMG  : :  ZG2 : 
MG2  : :  1  :  4,  .-.A  SMG  =  4  times  A  TZG=  6  times  area  AZG  = 
3  times  area  MAG  ;  hence  area  MAG  =i  A  SMG. 

(183.)  But  the  area  of  a  Parabola  may  be  ascertained  in  terms  of 
the  square  of  its  latus-rectum.  For  let  AN  :  AS  :  :  n  :  1,  then  AN= 
n  .  AS  ;  but    PN2  =  4AS  x  AN  =  An  .  AS2,    .-.  PN  =  2AS.  Vn  ; 

3 

A?i2 
hence  area  APN  (|  ANxPN)=  f  X  n.  AS  X  2AS  xVrc=-^-xAS2  = 


3, 

n2 


(for   AS=£BC,    and    .-.  AS2=TV  BC2V~  >cBC2;  or  if  the  whole 


3 

n2 


Parabola  is  taken  (as  in  Fig.  p.  28,)  then  the  area  MAG  =—  x  square 
of  latus-rectum. 

(184.)  Not  only  the  area  ANP  contained  between  the  abscissa 
and  ordinate,  but  also  the  area  ASP  described  by  the  revolution  of 
the  line  SP  round  the  focus  S,  may  be  ascertained  in  the  same  man- 
ner.    For  since  AN=n.AS,  SN  =  (AN— AS=)(rc— 1).AS ;  hence 

A  SPN  =  (iSN  X  PN  =  )n^-  .  AS  x  PN.      Now  area  APS  =  area 

APN— ASPN=fra.ASxPN ^-  xASxPN=  -y-.ASx 

PN=(for  PN  =  2AS Vn)  £+J^  xAS»-(-^±J^  xBC*. 


(185.)  Hence  it  appears,  that  if  the  latus-rectum  be  given,  the  par- 
abolic areas  ANP,  ASP  may  be  found  without  any  other  irrationality 
C.S.  17 


130 


QUADRATURE    OP   THE    ELLIPSE. 


than  that  which  arises  from  extracting  the  square  root  of  numbers  5 

for  iff*-* 1,  4,  9,  16,  &c.  then 

Area  ANP  =  fe  f,  f,  y,  &c.  of  the  square  of  the  latus-rectum  j  and 

Area  ASP  =  fa  fa  f,  if,  &c.  of  the  same ;  but  if  n  be  not  a  square 

number,  then  the  expression  for  these  areas  will  involve  an  irrational 

quantity. 


On  the  Quadrature  of  ^Ellipse. 


(186.)  Let  ABMO  be  an  Ellipse,  and  upon  the  major  axis  AM 
describe  the  circle  ARML  ;  draw  any  ordinate  Q,PN,  then  by  Prop- 
erty 9,  of  the  Ellipse,  Q,N  :  PN  in  the  given  ratio  of  RC  or  AC  :  BC. 
But  from  what  was  proved  in  Sect.  20,  area  AQN  :  area  APN  : : 
QN  :  PN  : :  AC  :  BC  ;  and  for  the  same  reason,  the  semicircle  ARM 
will  be  to  the  semi-ellipse  ABM  in  the  same  ratio ;  hence  the  whole 
Ellipse  ABM  :  circle  ARML  described  upon  its  major  axis  : :  BC  : 
AC  : :  minor  axis  :  major  axis. 


(187.)  As  the  area  of  the  Ellipse  bears  this  given  ratio  to  the 
area  of  its  circumscribing  circle,  the  quadrature  of  the  Ellipse 
must  therefore  depend  upon  the  quadrature  of  the  circle.     Let 


QUADRATURE   OF   THE    HYPERBOLA.  131 

p  =3.1416  (=*areaof  a  circle  whose  radius  is  1),  then  the  area 
of  the  circle  whose  radius  is  AC  -■  p  X  AC2 ;  hence  the  area  of  the 
Ellipse  :  /;xAC2  ::  BC  :  AC,  .-.area  of  the  Ellipse  =  p  x  AC  X 
BC,  i.  e.  the  area  of  an  Ellipse  is  found  by  multiplying  the  rectangle 
under  its  semi-axes  by  the  same  decimal  number  (p)  as  the  square 
of  the  radius  is  multiplied  by,  to  find  the  area  of  a  circle.  From 
this  it  also  appears,  that  the  area  of  an  Ellipse  is  equal  to  the  area 
of  a  circle  whose  radius  is  a  mean  proportional!  between  its  semi- 
axes  ;  for  the  area  of  that  circle  is  equal  to  (p  x(rad.)  2=p  xthe 
square  of  V  ( AC  x  BC)  -  )  p  x  AC  X  BC. 

(188.)  The  area  of  the  parallelogram  circumscribing  the  Ellipse 
is  equal  to  4 AC  x  BC,  .-.  area  of  Ellipse  :  area  of  that  parallelo- 
gram : :  p  x  AC  xBC  ;  4AC  xBC  : :  2h  or  3.1416  :  4  : :  .7854  :  1 ; 
i.  e.  the  area  of  an  Ellipse  has  the  same  ratio  to  the  area  of  its  cir- 
cumscribing parallelogram  as  the  area  of  a  circle  has  to  its  circum- 
scribing square. 


On  the  Quadrature  of  the  Hyperbola. 

(189.)  Let  AP/>  be  an  Hyperbola  whose  semi-axis  major  AC  =  a, 
semi-axis  minor  bC=b  ;  and  let  CN  =  .r,  PN  =  ?/;  then  by  Cor.  1. 
Prop.  6.  of  Hyperbola,  CN2— CA2  :  PN2  : :  AC2  :  BC2,  or   x%— 

a1  iifiia*:  62,  .-.y  =  - V(^2~«2);  hence  yx'  =  --.  x    V(^2— a2), 
a  a 

whose  fluent  found  by  a  series  and  properly  corrected  would  give 
the  value  of  the  area  APN ;  but  this  area  may  be  ascertained  by 
means  of  logarithms,  when  we  have  found  the  value  of  the  hyperbo- 
lic sector  ACP.     (See  Fig.  in  p.  126.) 

(190.)  Now  the  area  of  this  sector  is  thus  found.  The  area  of 
A  CPN  =  -£CN  xPN  =  %  .-.  the  fluxion  of  the  A  CPN  =  &p* . 


*  See  Day's  Mensuration,  &c,  Art.  30. 

t  Let  x  —  mean  proportional  between  AC  and  BC,  then  AC  : 
x  : :  x  :  BC,  .-.  x>  =  AC  x  BC,  or  x=  V (AC  x  BC.) 


132  QUADRATURE   OP   THE    HYPERBOLA. 

but    sector    ACP  =  A  PCN  —  area    APN,    .-.  fluxion   of  sector 
ACP  =  fluxion  of  A  PCN— fluxion  of  area  APN= — ^—  —  Vx  = 

xy~vx  ;  we  must  therefore  find  the  values  of  ~  and  y~.     Since  y  = 

b       ,    „         0\     •  bxx  xy  bx°- x  u         »  -a.       ion 

-V(^-a2)5,  =  ^-2-_z^,.^==5^7^2-_z^;    by    Art.    189. 


a^x*^a?f  '  ~2  ...  2a V (x^d 
■  ;  hence  xv  —  yx    or  ^ 

bx*x  bx  V  (#2 — a2)  aJa;' 


^=^V(f— };  hence  2=£   or  fluxion    of    sector    ACP  = 

2  2a  2 


the  fluent    or  sec- 


2aV(^2— a2)  2a  2v(x2— a2)' 

tor  ACP  =  u-  X  hyp.  log.  (x  -f  V  (#2 — a2)  )  +  Cor. ;  when  x  =  a, 

ACP  =  0,  ...  sector  ACP  =  $ x hyp.  log.  *  +  v(*'— g'). 

2  °  a 

(191.)  The  triangle  CPN=^=^^-2^^)J     .-.area    APN 

/&  /*a 

(=   A  CPN- sector   ACP)=  tov(^-g)_g»   x    hyp.    log. 

j-L-v  (#2 a2)  * 

— ' i. i.     Suppose  AQ,a  to  be  an  equilateral  hyperbola,  in 

which  a  =  6  =  l,  then  the  area  AQ,N=£rV(:r2 — 1) — £  hyp.  log. 
(x-{-V(z2— 1)  ).  A  portion  of  this  hyperbola,  whose  abscissa  is 
equal  to  its  semi-axis  major  (in  which  case  x  =  2)  will  be  numerical- 
ly expressed  by  the  quantity  V3—  \  hyp.  log.  (2-f  V3)  =  1.7320 — 
.6584=1.0736 ;  thus  in  Figure  page  82,  if  the  abscissa  AN  be  taken 
equal  to  AC,  then  the  area  (APN)  corresponding  to  this  abscissa : 
square  ACBa  : :  1.0736  :  1,  and  area  APN  :  quadrant  ACB  :  : 
1.0736  :  .7854  : :  1.3669  :  1. 


QUADRATURE    OF    THE    PARABOLA,    &C. 


133 


XXI. 

On  the  Quadrature  of  the  Parabola,  according  to  the  method  of 

the  Ancients. 

(192.)  Let  BQAPC  be  any  portion  of  a  Parabola  cut  off  by  the 
straight  line  BC ;  bisect  BC  in  the  point  D,  and  draw  DA  parallel 
to  the  axis ;  then  AD  will  be  the  diameter  to  the  point  A,  and  (by 
converse  of  Art.  23.)  BC  will  be  an  ordinate  to  that  diameter. 
Moreover,  since  a  tangent  to  the  point  A  is  parallel  to  BC,  A  will  be 
the  highest  point  or  vertex  of  the  figure  B&APC  ;  if  therefore  BA, 
AC,  be  joined,  then  this  figure  and  the  triangle  ABC  will  have  the 
same  base  and  vertex. 


(193.)  Bisect  BD  in  E,  and  draw  EQ,  parallel  to  DA  ;  through  Q, 
draw  GtNP  parallel  to  BC,  and  from  P  draw  PF  parallel  to  AD ; 
then  QNP  will  be  an  ordinate  to  the  diameter  AD  in  the  point  N, 
and  Q,E,  PF  will  be  diameters  to  the  points  Q,,  P  respectively ; 
and  since  Q,EDN  is  a  parallelogram,  Q,N  will  be  equal  to  ED,  i.  e. 
to  £BD;  hence*  QN2  :  BD2  : :  1  :  4 ;  but  by  the  property  of  the 
Parabola,  AN  :  AD  : :  QN2  :  BD2, .-.  AN  :  AD  : :  1  :  4,  or  AN  = 
£AD  ;  hence  ND  or  Q,E  =  f  AD.  Again,  since  EG  is  parallel  to 
DA,  and  BE  =  ^BD,  EG  must  be  equal  to  £AD,  .-.  Q.G  =  { f  AD,  and 
EG:  GQ,::  2:1. 

(194.)  Join  AE,  Ad,  GIB ;  then  since  BD  is  bisected  in  E,  the 
triangle  ABE  is  equal  to  half  the  triangle  ABD  (by  Euc.  1.  6. ;) 
and  since  GQ,  is  equal  to  £GE,  the  triangles  A&G,  BQ,G  are  res- 
pectively half  of  the  triangles  AGE,  BGE  ;  hence  the  triangle  AQ,B 
is  half  of  the  triangle  ABE,  and  consequently  ^th  of  the  triangle 
ABD.     In  the  same  manner  (if  AP,  PC,  AF,  be  joined,)  it  may 


134  QUADRATURE    OF   THE    PARABOLA. 

be  proved  that  the  triangle  APC  is  {th  the  triangle  ADC  ;  hence  the 
two  triangles  AQB,  APC,  taken  together,  are  equal  to  one  fourth  of 
the  triangle  ABC. 

(195.)  Now  suppose  BE,  ED  were  bisected,  and  from  the  points 
of  bisection  lines  were  drawn  parallel  to  DA  (which  will  evidently 
bisect  BG,  G/.,»)  then  the  sum  of  the  triangles  formed  within  the 
parabolic  spaces*'  BQ,  QA  (by  drawing  lines  from  the  points  where 
those  parallel  lines  cut  the  curve  to  the  extremities  of  the  chords  BQ, 
QA)  will  be  equal  to  ]th  of  the  triangle  AQBt ;  and  the  sum  of  the 
triangles  formed  in  a  similar  manner  within  the  parabolic  spaces  AP, 
PC,  will  be  equal  to  {th  of  the  triangle  APC  ;  .-.  the  sum  of  the  tri- 
angles formed  within  the  four  parabolic  spaces  BQ,,  QA,  AP,  PC  is 
equal  to  ith  of  A  AQB+Z^VPC,  i.  e.  to  Tyth  of  the  triangle  ABC. 
By  bisecting  the  halves  of  BE,  ED,  &c.  and  drawing  lines  as  be- 
fore, parallel  to  DA,  and  joining  the  points  of  their  intersection  with 
the  curve  to  the  extremities  of  the  chords,  a  series  of  eight  triangles 
would  be  formed  in  the  remaining  parabolic  spaces,  the  sum  of  Which 
would  be  equal  to  {th  of  the  sum  of  the  triangles  formed  within  the 
parabolic  spaces  BQ,  QA,  AP,  PC,  i.  e.  to  T*Tth  of  the  triangle 
ABC.  We  might  thus  go  on  bisecting  the  successive  parts  of  the 
base  BC,  and  forming  triangles  in  a  similar  manner,  till  the  whole 
parabolic  figure  BAC  was  exhausted,  in  which  case  it  is  evident  that 
the  area  of  that  figure  would  be  equal  to  the  sum  of  the  areas  of  all 
the  triangles  thus  formed  within  it. 

(196.)  Let  the  triangle  ABC  =  a,  then  to  find  the  sum  of  the 
areas  of   all   these   triangles,   we  have  merely  to  sum  the  series 


*  By  parabolic  spaces,  we  mean  such  portions  of  the  Parabola  as 
are  contained  between  the  arcs  BQ,  QA.  AP,  PC,  and  the  straight 
lines  BQ,  QA,  AP,  PC  respectively. 

t  For  the  same  reason  that  the  sum  of  the  triangles  AQB,  APC 
is  equal  to  ith  the  triangle  ABC,  this  conclusion  being  evidently  true 
for  the  triangles  thus  inscribed  in  any  portion  of  a  Parabola. 


BY  THE  METHOD  OF  THE  ANCIENTS.  135 

a_|_a_j-iL_|-iL -f&c.  continued  ad  infinitum,  which  is  a  geometric 

series,  whose  first  term  is  a,  and  common  ratio  J,     Now  the  sum  of 

this  series*  =  1  7— =li =-tt  5    •'•    tne    area    °f    tne    parabola 

VI— r    /l—\     a 

BAC  is  equal  to  f  x  area  of  the  A  BAC.  If  a  tangent  was  drawn 
to  the  point  A,  and  from  B,  C,  lines  were  drawn  parallel  to  DA,  then 
the  triangle  ABC  would  be  the  half  of  the  parallelogram  thus  formed  ; 
the  parabolic  area  BAC  is  therefore  fds  of  the  circumscribing  paral- 
lelogram ;  which  accords  with  what  has  already  been  proved 
respecting  the  quadrature  of  the  Parabola  in  Section  XX ;  for  it  is 
evident  the  foregoing  demonstration  is  true  for  the  axis,  since  AD  is 
any  diameter. 

(197.)  From  the  given  ratio  which  subsists  between  the  parabolic 
area  and  its  inscribed  triangle,  we  may  prove,  that  such  portions  of 
a  Parabola  as  are  cut  oif  by  ordinates  to  equal  diameters,  are  equal  to 
one  another.  Let  oAQ  (Fig.  in  p.  136)  be  any  Parabola,  and  draw 
the  diameters  P  W,  pw  to  the  points  P,  p  ;  take  PW=p«?,  and  through 
W,  w,  draw  the  ordinates  OWQ,  owq  ;  draw  the  axis  AD  ;  take  AD 
equal  to  PW  or  pw,  and  through  D  draw  the  ordinate  BC ;  and  in 
the  parabolic  spaces  BAC,  OPQ,  inscribe  the  triangles  BAC,  OPQ. 
Draw  the  tangent  to  the  point  P,  and  produce  the  axis  to  meet  it  in 
the  point  T  ;  let  S  be  the  focus,  and  join  SP ;  from.S  let  fall  SY 
perpendicular  upon  the  tangent,  and  draw  QF  perpendicular  upon 
PW  produced.  Now  4SAxAD  =  CD2,  and  4SPxPW=WQ,2; 
therefore  WQ2  :  CD2  ::  4SPxPW  :  4SAxAD  : :  (since  PW= 
AD)  SP  :  SA.  Again,  since  the  ordinate  WQ.  is  parallel  to  the  tan- 
gent TP,  and  the  diameter  PW  is  parallel  to  the  axis  AD,  the  tri- 
angles WQF,  STY  are  similar,  .-.  WQ2  :  QF2  j :  ST2  (or  SP2)  : 
SY2  : :  (Euc.  Def.  11.  5.)  SP  :  SA ;  hence  WQ2  :  CD2  : :  WQ2  : 
QF2,  .-.  CD=QF.  But  the  A  PWQ=iPWxQF  and  the  A 
ADC=JADxCD;  since  therefore  PW,  QF,  are  respectively 
equal  to  AD,  CD,  the   A  PWQ  must  be  equal  to  the  A  ADC. 


*  See  Day's  Algebra,  Art.  442. 


136  QUADRATURE    OF   THE    PARABOLA,    &C. 

T 


Now  these  A  s  are  the  halves  of  the  triangles  OPQ  and  ABC ; 
hence  the  A  OPQ,  is  equal  to  the  triangle  ABC,  and  consequently 
the  Parabolic  area  OPQ,  to  the  parabolic  area  BAC*  In  the  same 
manner  it  might  be  proved  that  the  parabolic  area  opq  is  equal  to 
the  area  BAC  ;  .-.  the  area  opq  is  equal  to  the  area  OPQ. 

These  observations  upon  the  quadrature  of  the  Parabola  accord- 
ing to  the  method  of  the  Ancients,  contain  the  substance  of  the  last 
seven  propositions  (viz.  from  18  to  24  inclusive)  of  Archimedes  De 
Quadratures  Parabola,  and  of  the  fourth  Proposition  of  his  book 
De  Conoidibus  et  Sphceroidibus. 


*  For  by  Art.  195,  the  parabolic  areas  OPQ,  BAC  are  f  ds  of  the 
triangles  OPQ,  BAC  respectively. 


THE    END. 


Or    THfc 


"hut 


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6Jun'55AM 


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MAR  24  1959 


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\H-5ZHO 


THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


